570 lines
24 KiB
Markdown
570 lines
24 KiB
Markdown
---
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title: Combinations
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TARGET DECK: Obsidian::STEM
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FILE TAGS: combinatorics set
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tags:
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- combinatorics
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- set
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---
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## Overview
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A $k$-**combination** of $n$ objects is an unordered "choice" of $k$ objects from the collection of $n$ objects. Alternatively viewed, it is a set of $k$ objects - ordering within a set does not matter. Combinations are derived by considering the number of $k$-[[permutations]] of $n$ objects and discarding order, i.e. dividing by $k!$. $$\binom{n}{k} = \frac{(n)_k}{k!} = \frac{n!}{k!(n - k)!}$$
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```c
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void combinations_aux(
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const size_t n, int A[static n],
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const size_t k, int stack[static k],
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const size_t i
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) {
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if (n < k) {
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return;
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}
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if (k == 0) {
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print_array(i, stack);
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return;
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}
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stack[i] = A[0];
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combinations_aux(n - 1, A + 1, k - 1, stack, i + 1);
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combinations_aux(n - 1, A + 1, k, stack, i);
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}
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void combinations(const size_t n, const size_t k, int A[static n]) {
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int *stack = calloc(k, sizeof(int));
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combinations_aux(n, A, k, stack, 0);
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free(stack);
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}
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```
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The above approach prints out all $k$-combinations of an array.
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%%ANKI
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Basic
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What *is* a combination?
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Back: An unordered collection of objects.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078673-->
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END%%
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%%ANKI
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Cloze
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{1:Permutations} are to {2:tuples} as {2:combinations} are to {1:sets}.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078679-->
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END%%
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%%ANKI
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Basic
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How is a $k$-combination of $n$ objects denoted?
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Back: $\binom{n}{k}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078685-->
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END%%
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%%ANKI
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Basic
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How is $\binom{n}{k}$ pronounced?
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Back: "$n$ choose $k$"
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078690-->
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END%%
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%%ANKI
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Basic
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How is $\binom{n}{k}$ combinations of $n$ objects derived?
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Back: As $(n)_k$ $k$-permutations of $n$ divided by $k!$, the numer of possible $k$-orderings.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078695-->
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END%%
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%%ANKI
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Basic
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Without using falling factorials, what is the closed formula of $\binom{n}{k}$?
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Back: $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078700-->
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END%%
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%%ANKI
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Basic
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Using falling factorials, what is the closed formula of $\binom{n}{k}$?
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Back: $$\binom{n}{k} = \frac{(n)_k}{k!}$$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708532069694-->
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END%%
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%%ANKI
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Basic
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How do $k$-permutations of $n$ objects relate to $k$-combinations?
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Back: The number of $k$-combinations is the number of $k$-permutations divided by $k!$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078705-->
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END%%
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%%ANKI
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Basic
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How is the closed formula of $\binom{n}{k}$ written in terms of falling factorials?
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Back: $$\binom{n}{k} = \frac{(n)_k}{k!}$$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078711-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept explains the number of subsets of a finite set of a given size?
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Back: Combinations.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078717-->
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END%%
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%%ANKI
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Basic
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How many subsets of $\{a, b, c, d, e\}$ have exactly $3$ members?
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Back: $\binom{5}{3}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078723-->
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END%%
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%%ANKI
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Basic
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*Why* are binomial coefficients "symmetric"?
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Back: The number of ways to choose $k$ objects is the same as the number of ways to *not* choose those $k$ objects.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078729-->
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END%%
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%%ANKI
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Basic
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What value of $k \neq 1$ makes $\binom{n}{1} = \binom{n}{k}$?
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Back: $n - 1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078735-->
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END%%
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%%ANKI
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Basic
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What does $\binom{n}{0}$ evaluate to?
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Back: $1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078740-->
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END%%
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%%ANKI
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Basic
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What does $\binom{n}{n}$ evaluate to?
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Back: $1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078746-->
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END%%
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%%ANKI
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Basic
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How many *increasing* injective functions exist between $\{1, 2, 3\}$ and $\{a, b, c, d, e\}$?
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Back: $\binom{5}{3}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708446818783-->
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END%%
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%%ANKI
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Basic
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How many *decreasing* injective functions exist between $\{1, 2\}$ and $\{a, b, c, d\}$?
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Back: $\binom{4}{2}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708446818786-->
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END%%
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%%ANKI
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Basic
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Given finite sets $A$ and $B$, what is the number of *increasing* injective functions between $A$ and $B$?
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Back: Given $k = |A|$ and $n = |B|$, $\binom{n}{k}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708446818788-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept explains the number of *increasing* injective functions between two finite sets?
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Back: Combinations.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708446818789-->
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END%%
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%%ANKI
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Basic
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Given $k = |A|$ and $n = |B|$, *why* is the number of increasing injective functions between $A$ and $B$ equal to $\binom{n}{k}$?
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Back: We are "grouping" all functions by a shared permutation (i.e. the increasing function).
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708446818791-->
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END%%
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## Pascal's Triangle
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A visual representation of the binomial coefficient's is in the form of Pascal's Triangle:
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```
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1
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1 1
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1 2 1
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1 3 3 1
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1 4 6 4 1
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1 5 10 10 5 1
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...
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```
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Terms are generated by adding the two terms above it, formalized via recurrence $$\binom{n}{k} = \binom{n - 1}{k} + \binom{n - 1}{k - 1}$$
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%%ANKI
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Basic
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What recurrence relation does $\binom{n}{k}$ satisfy?
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Back: $$\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441360-->
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END%%
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%%ANKI
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Basic
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*Why* is it that $\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$?
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Back: For each member, we either include in a subset or we don't.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708446818792-->
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END%%
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%%ANKI
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Basic
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What name is given to the following structure?
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```
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1
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1 1
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1 2 1
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1 3 3 1
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1 4 6 4 1
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```
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Back: Pascal's Triangle.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441393-->
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END%%
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%%ANKI
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Basic
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What recurrence relation is Pascal's Triangle based on?
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Back: $$\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441399-->
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END%%
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%%ANKI
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Basic
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What binomial coefficient is at the peak of Pascal's Triangle?
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Back: $\binom{0}{0}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441405-->
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END%%
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%%ANKI
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Basic
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What is the value of $\sum_{k=0}^n \binom{n}{k}$?
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Back: $2^n$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441412-->
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END%%
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%%ANKI
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Basic
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What is the sum of Pascal's Triangle's $n$th row?
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Back: $2^n$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441417-->
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END%%
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%%ANKI
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Basic
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Why does $\sum_{k=0}^n \binom{n}{k} = 2^n$?
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Back: The summands count the number of subsets of a set of cardinality $n$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708384441423-->
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END%%
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## Bit Strings
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A [[binary/index|bit string]] can be used to represent subsets of some finite set. A `1` value usually corresponds to inclusion in a subset, whereas a `0` value corresponds to exclusion. Thus, given set e.g. $A = \{1, 2, 3, 4\}$, $0110_2$ would correspond to subset $\{2, 3\}$.
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%%ANKI
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Basic
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Given $A = \{1, 2, 3, 4\}$, what subset does $1010_2$ correspond to?
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Back: $\{1, 3\}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662140-->
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END%%
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%%ANKI
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Cloze
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Representing subsets with bit strings, {$0000_2$} maps to {$\varnothing$} $\subseteq \{1, 2, 3, 4\}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662148-->
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END%%
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%%ANKI
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Cloze
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Representing subsets with bit strings, {$1111_2$} maps to {$\{1, 2, 3, 4\}$} $\subseteq \{1, 2, 3, 4\}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662151-->
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END%%
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Bit strings also make it clear that the number of subsets with even cardinality must be equal to the number of subsets with odd cardinality. Hence, $$\binom{n}{0} - \binom{n}{1} + \cdots + (-1)^n \binom{n}{n} = 0$$
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%%ANKI
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Basic
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What does $\sum_{k=0}^n (-1)^k \binom{n}{k}$ evaluate to?
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Back: $0$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662154-->
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END%%
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%%ANKI
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Basic
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What does expression $\sum_{k=0}^n (-1)^k \binom{n}{k} = 0$ indicate about sets?
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Back: There is an equal number of even-sized subsets as odd-sized subsets.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662157-->
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END%%
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%%ANKI
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Basic
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What isomorphism shows finite sets have an equal number of even-sized subsets as odd-sized subsets?
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Back: The one-to-one relationship between subsets and bit strings.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662160-->
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END%%
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%%ANKI
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Basic
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Given finite set $A$, how many subsets of even cardinality does $A$ have?
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Back: Let $n = |A|$. Then $2^{n-1}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662163-->
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END%%
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%%ANKI
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Basic
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Given finite set $A$, how many subsets of odd cardinality does $A$ have?
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Back: Let $n = |A|$. Then $2^{n-1}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708434662166-->
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END%%
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### Stars and Bars
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The **stars and bars** chart refers to a graphical depiction of distributing $n$ objects (represented as $*$) into $m$ different buckets (delineated via $|$. An example chart looks like so: $$**|***|*||*$$
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Notice there are $m - 1$ bars and interspersed amongst the $n$ stars. In the above example, there are $11$ total symbols, $4$ of which are bars, meaning there are $\binom{11}{4}$ ways to distribute the objects amongst the $5$ buckets. We can represent this using bit strings instead, with `0`s as stars and `1`s as bars. The above example is equivalently written as: $$00100010110$$
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%%ANKI
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Basic
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What symbols are typically used in a stars and bars chart?
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Back: $*$ and $|$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708532069700-->
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END%%
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%%ANKI
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Basic
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What property is exhibited by objects distributed in a stars and bars chart?
|
|
Back: They are identical to one another.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708532069704-->
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|
END%%
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|
%%ANKI
|
|
Basic
|
|
What kind of chart is the following an example of? $$**|***|*||*$$
|
|
Back: Stars and bars
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708532069707-->
|
|
END%%
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|
%%ANKI
|
|
Basic
|
|
How is the following stars and bars chart written as a bit string? $$**|***|*||*$$
|
|
Back: $00100010110$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708532069711-->
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|
END%%
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|
%%ANKI
|
|
Basic
|
|
The following stars and bars chart is a single instance of how many possible choices? $$**|***|*||*$$
|
|
Back: $\binom{11}{4}$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708532069714-->
|
|
END%%
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|
%%ANKI
|
|
Basic
|
|
What graphical depiction is used to find the number of integer solutions to the following equation? $$x_1 + x_2 + \cdots + x_k = n$$
|
|
Back: Stars and bars
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708532069717-->
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|
END%%
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|
%%ANKI
|
|
Basic
|
|
What distribution does the following stars and bars chart represent? $$*||*$$
|
|
Back: A single object in the first and last bucket. No object in the middle.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708532069719-->
|
|
END%%
|
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|
|
## Lattice Paths
|
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|
|
A **lattice path** is one of the shorted possible paths connecting two points on a lattice, moving only horizontally and vertically. By representing each horizontal move by `1` and each vertical move by `1`, we see every lattice path has a corresponding [[#Bit Strings|bit string]].
|
|
|
|
![[lattice-path-example.png]]
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|
|
|
In this example, the total number of lattice paths from point $(0, 0)$ to $(3, 2)$ is therefore $\binom{5}{2} = \binom{5}{3}$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many lattice paths are there from $(0, 0)$ to $(n, n)$?
|
|
Back: $\binom{2n}{n}$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708451749788-->
|
|
END%%
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|
|
|
%%ANKI
|
|
Basic
|
|
What is the integer lattice?
|
|
Back: $\omega \times \omega = \{(x, y) : x \in \omega \land y \in \omega\}$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384441429-->
|
|
END%%
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|
|
|
%%ANKI
|
|
Basic
|
|
What is a lattice path?
|
|
Back: A possible shortest path between two points on an integer lattice.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384441434-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many lattice paths exist from one corner of a chess board to the opposite corner?
|
|
Back: $\binom{14}{7}$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384441440-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many horizontal moves must a lattice path make between $(0, 0)$ and $(3, 2)$?
|
|
Back: $3$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384507600-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many vertical moves must a lattice path make between $(0, 0)$ and $(3, 2)$?
|
|
Back: $2$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384507605-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many moves (horizontal or vertical) must a lattice path make between $(0, 0)$ and $(3, 2)$?
|
|
Back: $5$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384507608-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How do lattice paths relate to bit strings?
|
|
Back: Denoting horizontal and vertical moves by $H$ and $V$ makes the relationship clear.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384507611-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many lattice paths exist between $(0, 0)$ and $(3, 2)$?
|
|
Back: $\binom{5}{2} = \binom{5}{3}$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384507615-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the number of lattice paths in the following shown to mirror the binomial recurrence?
|
|
![[lattice-path-before-recurrence.png]]
|
|
Back:
|
|
It equals the number of lattice paths to $A$ plus the number of lattice paths to $B$.
|
|
![[lattice-path-after-recurrence.png]]
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384507618-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The number of lattice paths to the marked square can be derived by adding the values of which two squares?
|
|
![[lattice-path-chessboard.png]]
|
|
Back:
|
|
![[lattice-path-chessboard-colored.png]]
|
|
<!--ID: 1708384441446-->
|
|
END%%
|
|
|
|
## Binomial Coefficients
|
|
|
|
A **binomial** is a polynomial containing two terms. Consider $(x + y)^n$. We see that term $x^ky^{n-k}$ maps to *some* bit string containing $k$ `1`s and $n - k$ `0`s. This might feel more obvious when expanding to $x \cdot x \cdots x \cdot y \cdots y$. Since multiplication is commutative, the number of matching "bit strings" is the same as $\binom{n}{k}$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What term describes e.g. $\binom{n}{1}$, $\binom{n}{2}$, etc.?
|
|
Back: The binomial coefficients.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708368078753-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Why are binomial coefficients named the way they are?
|
|
Back: They are the coefficients of terms in the expansion of a binomial, e.g. $(x + y)^n$.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708368078764-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How does $(x + y)^n$ relate to bit strings?
|
|
Back: Each term $x^ky^{n-k}$ corresponds to a bit string with $k$ `1`s and $n - k$ `0`s.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384441451-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the coefficient of $x^3y^{12}$ in $(x + y)^{15}$?
|
|
Back: $\binom{15}{3} = \binom{15}{12}$.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384441456-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $(x + y)^n$ written to mirror the binomial recurrence?
|
|
Back: $(x + y)^n = x \cdot (x + y)^{n-1} + y \cdot (x + y)^{n - 1}$
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708384441462-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $7^n$ written as a sum of powers of $6$?
|
|
Back: $7^n = (1 + 6)^n$. Apply binomial expansion on the RHS.
|
|
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
|
<!--ID: 1708451749791-->
|
|
END%%
|
|
|
|
## Bibliography
|
|
|
|
* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). |