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title TARGET DECK FILE TAGS tags
Combinations Obsidian::STEM combinatorics set
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%%

%%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 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.

END%%

References