15 KiB
title | TARGET DECK | FILE TAGS | tags | ||
---|---|---|---|---|---|
Combinations | Obsidian::STEM | combinatorics set |
|
Overview
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)!}
void combinations_aux(
const int i, const int n,
const int j, const int k,
int A[static n],
int res[static k]
) {
if (j == k) {
for (int m = 0; m < k; ++m) {
printf("%d ", A[res[m]]);
}
printf("\n");
return
} else if (n - i < k - j) {
return;
}
res[j] = A[i];
combinations_aux(i + 1, n, j + 1, k, A, res);
combinations_aux(i + 1, n, j, k, A, res);
}
void combinations(const int n, const int k, int A[static n]) {
int *res = malloc(sizeof(int) * k);
combinations_aux(0, n, 0, k, A, res);
free(res);
}
The above approach prints out all k
-combinations of a given array.
%%ANKI Basic What is a combination? Back: An unordered collection of objects. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze {1:Permutations} are to {2:tuples} as {2:combinations} are to {1:sets}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is a k
-combination expressed recursively?
Back: Include or exclude a candidate, then find (k - 1)
- or k
-combinations on the remainder.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is a k
-combination of n
objects denoted?
Back: \binom{n}{k}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is \binom{n}{k}
pronounced?
Back: "n
choose k
"
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is \binom{n}{k}
combinations of n
objects derived?
Back: As (n)_k
k
-permutations of n
divided by k!
, the numer of possible k
-orderings.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is the closed formula of \binom{n}{k}
written in terms of factorials (not falling factorials)?
Back: \binom{n}{k} = \frac{n!}{k!(n - k)!}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How do k
-permutations of n
objects relate to k
-combinations?
Back: The number of k
-combinations is the number of k
-permutations divided by k!
.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is the closed formula of \binom{n}{k}
written in terms of falling factorials?
Back: \binom{n}{k} = \frac{(n)_k}{k!}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What combinatorial concept explains the number of subsets of a finite set? Back: Combinations. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How many subsets of \{a, b, c, d, e\}
have exactly 3
members?
Back: \binom{5}{3}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why are binomial coefficients "symmetric"
Back: The number of ways to choose k
objects is the same as the number of ways to not choose those k
objects.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What value of k \neq 1
makes \binom{n}{1} = \binom{n}{k}
?
Back: n - 1
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does \binom{n}{0}
evaluate to?
Back: 1
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does \binom{n}{n}
evaluate to?
Back: 1
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Pascal's Triangle
A visual representation of the binomial coefficient's is in the form of Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
...
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}
%%ANKI
Basic
What recurrence relation does \binom{n}{k}
satisfy?
Back: \binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What name is given to the following structure?
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Back: Pascal's Triangle. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What recurrence relation is Pascal's Triangle based on?
Back: \binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What binomial coefficient is at the peak of Pascal's Triangle?
Back: \binom{0}{0}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the value of \sum_{k=0}^n \binom{n}{k}
?
Back: 2^n
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the sum of Pascal's Triangle's $n$th row?
Back: 2^n
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why does \sum_{k=0}^n \binom{n}{k} = 2^n
?
Back: The summands count the number of subsets of a set of cardinality n
.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Lattice Paths
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 "0", we see every lattice path has a corresponding bit string.
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
What is the integer lattice?
Back: \{(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.
END%%
%%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.
END%%
%%ANKI
Basic
How many lattice paths exist from one corner of a chess board to the opposite corner?
Back: \binom{16}{8}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
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.
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.
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.
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.
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.
END%%
%%ANKI
Basic
How is the number of lattice paths in the following shown to mirror the binomial recurrence?
!
Back:
It equals the number of lattice paths to A
plus the number of lattice paths to B
.
!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic The number of lattice paths to the marked square can be derived by adding the values of which two squares? ! Back: !
END%%
Binomials
A binomial is a polynomial containing two terms. Consider (x + y)^n
.
%%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.
END%%
%%ANKI Basic What is a binomial? Back: A polynomial containing two terms. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
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.
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.
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.
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.
END%%
References
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.