Notes on permutations and combinations.

2024-02-19 12:13:11 -07:00 · 2024-02-19 12:13:11 -07:00 · 4fe3a60534
parent 2a2d8f8195
commit 4fe3a60534
7 changed files with 491 additions and 6 deletions
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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@ -124,7 +124,7 @@
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+    "combinatorics/permutations.md": "3c76f548e22f7271280b9ed82ae584ec",
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  "fields_dict": {
    "Basic": [
--- a/notes/_journal/2024-02-19.md
+++ b/notes/_journal/2024-02-19.md
@ -8,4 +8,6 @@ title: "2024-02-19"
 - [ ] OGS (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)
 - [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
+- [ ] Log Work Hours (Max 3 hours)
+
+* Start encoding notes on permutations and combinations, as well as their relationship to one another.
--- a/notes/binary/index.md
+++ b/notes/binary/index.md
@ -6,6 +6,46 @@ tags:
  - binary
 ---

+## Overview
+
+A binary digit or **bit** is a `0` or `1` character. A bit string is then a contiguous sequence of bits. It's **weight** is a reference to the number of `1`s in the bit string. Compare the below operation to the method for converting from one numerical base to another (e.g. [[hexadecimal]]).
+
+```c
+unsigned int bit_weight(int64_t n) {
+  unsigned int count = 0;
+  while (n) {
+    count += (n % 2 == 0) ? 0 : 1;
+    n /= 2;
+  }
+  return count;
+}
+```
+
+%%ANKI
+Basic
+Why is a "bit" named the way it is?
+Back: It is short for **b**inary dig**it**.
+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: 1707432641557-->
+END%%
+
+%%ANKI
+Basic
+What does the weight of a bit string refer to?
+Back: The number of `1`s in the string.
+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: 1708366788645-->
+END%%
+
+%%ANKI
+Basic
+How might you use C to find the weight of a bit string?
+Back: Repeatedly divide by `2`, counting all remainders of `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).
+Tags: c
+<!--ID: 1708366788648-->
+END%%
+
 %%ANKI
 Cloze
 A byte consists of {8} bits.
--- a/notes/combinatorics/additive-principle.md
+++ b/notes/combinatorics/additive-principle.md
@ -4,12 +4,15 @@ TARGET DECK: Obsidian::STEM
 FILE TAGS: combinatorics set
 tags:
  - combinatorics
+  - set
 ---

 ## Overview

 The **additive principle** states that two finite and disjoint sets $A$ and $B$ satisfy $$|A \cup B| = |A| + |B|$$

+This can be generalized to any number of finite and disjoint sets in the obvious way.
+
 %%ANKI
 Basic
 What does the additive principle state?
--- a/notes/combinatorics/combinations.md
+++ b/notes/combinatorics/combinations.md
@ -0,0 +1,189 @@
+---
+title: Combinations
+TARGET DECK: Obsidian::STEM
+FILE TAGS: combinatorics set
+tags:
+  - 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 $k!$. $$\binom{n}{k} = \frac{(n)_k}{k!} = \frac{n!}{k!(n - k)!}$$
+
+```c
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078673-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078679-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708369553051-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078685-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078690-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078695-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078700-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078705-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078711-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078717-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078723-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078729-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078735-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078740-->
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078746-->
+END%%
+
+%%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
+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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708368078759-->
+END%%
+
+%%ANKI
+Basic
+What are binomial coefficients?
+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%%
+
+## References
+
+* 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).
--- a/notes/combinatorics/multiplicative-principle.md
+++ b/notes/combinatorics/multiplicative-principle.md
@ -11,6 +11,8 @@ tags:

 The **multiplicative principle** states that two finite sets $A$ and $B$ satisfy $$|A \times B| = |A| \cdot |B|$$

+This can be generalized to any number of finite sets in the obvious way.
+
 %%ANKI
 Basic
 What does the multiplicative principle state?
--- a/notes/combinatorics/permutations.md
+++ b/notes/combinatorics/permutations.md
@ -0,0 +1,247 @@
+---
+title: Permutations
+TARGET DECK: Obsidian::STEM
+FILE TAGS: combinatorics set
+tags:
+  - combinatorics
+  - set
+---
+
+## Overview
+
+A **permutation** of some $n$ objects is a (possible) rearrangement of those $n$ objects. The number of permutations is $n!$ since there are $n$ possible ways to pick the first object, $(n - 1)$ possible ways to pick the second, and so on.
+
+```c
+void permutations_aux(
+  const size_t n,
+  int A[static n],
+  int res[static n],
+  uint64_t choices
+) {
+  if (!choices) {
+    for (size_t i = 0; i < n; ++i) {
+      printf("%d ", A[res[i]]);
+    }
+    printf("\n");
+    return;
+  }
+
+  unsigned int weight = n - bit_weight(choices);
+  for (unsigned int i = 0; i < 64; ++i) {
+    uint64_t next = 1L << i;
+    if (choices & next) {
+      res[weight] = i;
+      permutations_aux(n, A, res, choices & ~next);
+    }
+  }
+}
+
+void permutations(const size_t n, int A[static n]) {
+  int *res = malloc(sizeof(int) * n);
+  permutations_aux(n, A, res, (1L << n) - 1);
+  free(res);
+}
+```
+
+The above approach prints out all permutations of a given array, provided the array contains at most `64` digits. It relies on `bit_weight` as defined in [[binary/index|binary]].
+
+%%ANKI
+Basic
+What *is* a permutation?
+Back: An ordered arrangement of some collection of objects.
+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: 1708366788567-->
+END%%
+
+%%ANKI
+Basic
+How many permutations are there of $n$ objects?
+Back: $n!$
+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: 1708366788573-->
+END%%
+
+%%ANKI
+Basic
+How is permutation expressed recursively?
+Back: Put each candidate in the current position, finding all permutations of the remainder each time.
+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: 1708369553046-->
+END%%
+
+%%ANKI
+Basic
+How is $n!$ permutations of $n$ objects derived?
+Back: There are $n$ choices for the first position, $n - 1$ choices for the second, etc.
+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: 1708366788576-->
+END%%
+
+%%ANKI
+Basic
+What combinatorial concept explains $n!$ permutations of $n$ objects?
+Back: The multiplicative principle.
+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: 1708366788580-->
+END%%
+
+%%ANKI
+Basic
+What combinatorial concept is often associated with the factorial?
+Back: Permutations.
+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: 1708366788583-->
+END%%
+
+%%ANKI
+Basic
+How does sorting relate to the concept of permutations?
+Back: Sorting aims to efficiently find a specific permutation.
+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).
+Tags: algorithm
+<!--ID: 1708366788587-->
+END%%
+
+%%ANKI
+Basic
+What symbol denotes "$n$ factorial"?
+Back: $n!$
+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: 1708366788590-->
+END%%
+
+%%ANKI
+Basic
+What is $n!$ shorthand for?
+Back: $\Pi_{k=1}^n k$
+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: 1708366788594-->
+END%%
+
+%%ANKI
+Basic
+What is the identity element of $\cdot$ (multiplication)?
+Back: $1$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1708366788597-->
+END%%
+
+%%ANKI
+Basic
+What does $0!$ (factorial) evaluate to?
+Back: $1$
+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: 1708366788600-->
+END%%
+
+%%ANKI
+Basic
+*Why* might $0! = 1$ (barring convention)?
+Back: Because the empty product is $1$, the multiplication identity.
+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: 1708366788603-->
+END%%
+
+%%ANKI
+Basic
+What combinatorial concept explains the number of bijective functions between two finite sets?
+Back: Permutations (factorials).
+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: 1708366788606-->
+END%%
+
+%%ANKI
+Basic
+How many bijective functions exist between $\{1, 2, 3\}$ and $\{a, b, c\}$?
+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: 1708366788610-->
+END%%
+
+%%ANKI
+Basic
+How many bijective functions exist between finite sets $A$ and $B$ where $|A| = |B| = n$?
+Back: $n!$
+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: 1708366788613-->
+END%%
+
+If we generalize to choosing $k \leq n$ elements of $k$ objects, we can calculate the $k$-permutation of $n$. This is denoted as $(n)_k$, sometimes called the **falling factorial**. $$(n)_k = \frac{n!}{(n - k)!}$$
+
+The derivation works by noting that we have $n - 0$ possible ways to pick the first object, $n - 1$ ways to pick the second, up until $n - (k - 1)$ ways to pick the last object.
+
+%%ANKI
+Basic
+What *is* a $k$-permutation?
+Back: An ordered arrangement, containing $k$ elements, of some collection of objects.
+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: 1708366788616-->
+END%%
+
+%%ANKI
+Basic
+What is the closed formula for $(n)_k$ (falling factorial)?
+Back: $$(n)_k = \frac{n!}{(n - k)!}$$
+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: 1708366788619-->
+END%%
+
+%%ANKI
+Basic
+How many $k$-permutations are there of $n$ objects?
+Back: $(n)_k$
+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: 1708366788622-->
+END%%
+
+%%ANKI
+Basic
+How is $(n)_k$ $k$-permutations of $n$ objects derived?
+Back: There are $n$ choices for the first position, $n - 1$ choices for the second, etc. up until $n - (k - 1)$ choices for the last position.
+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: 1708366788625-->
+END%%
+
+%%ANKI
+Basic
+What does $(n)_n$ evaluate to?
+Back: $n!$
+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: 1708366788628-->
+END%%
+
+%%ANKI
+Basic
+What does $(n)_0$ evaluate to?
+Back: $1$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1708366788631-->
+END%%
+
+%%ANKI
+Cloze
+In a $k$-permutation of $n$ objects, there are $n - 0$ choices for first object and {$n - (k - 1)$} choices for the last object.
+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: 1708366788634-->
+END%%
+
+%%ANKI
+Basic
+What combinatorial concept explains the number of injective functions between two finite sets?
+Back: $k$-permutations (falling factorials).
+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: 1708366788638-->
+END%%
+
+%%ANKI
+Basic
+How many injective functions exist between $\{1, 2, 3\}$ and $\{a, b, c, d, e\}$?
+Back: $(5)_3$
+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: 1708366788641-->
+END%%
+
+## References
+
+* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+* 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).