notebook/notes/combinatorics/combinations.md

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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 size_t n, int A[static n],
  const size_t k, int stack[static k],
  const size_t i
) {
  if (n < k) {
    return;
  }
  if (k == 0) {
    print_array(i, stack);
    return;
  }
  stack[i] = A[0];
  combinations_aux(n - 1, A + 1, k - 1, stack, i + 1);
  combinations_aux(n - 1, A + 1, k, stack, i);
}

void combinations(const size_t n, const size_t k, int A[static n]) {
  int *stack = calloc(k, sizeof(int));
  combinations_aux(n, A, k, stack, 0);
  free(stack);
}

The above approach prints out all k-combinations of an 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 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 Without using falling factorials, what is the closed formula of \binom{n}{k}? 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 Using falling factorials, what is the closed formula of \binom{n}{k}? 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 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 How many increasing injective functions exist between \{1, 2, 3\} and \{a, b, c, d, e\}? 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 How many decreasing injective functions exist between \{1, 2\} and \{a, b, c, d\}? Back: \binom{4}{2} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given finite sets A and B, what is the number of increasing injective functions between A and B? Back: Given k = |A| and n = |B|, \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 What combinatorial concept explains the number of increasing injective functions between two finite sets? 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 Given k = |A| and n = |B|, why is the number of increasing injective functions between A and B equal to \binom{n}{k}? Back: We are "grouping" all functions by a shared permutation (i.e. the increasing function). 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 Why is it that \binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}? Back: For each member, we either include in a subset or we don't. 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%%

Bit Strings

A binary/index 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\}.

%%ANKI Basic Given A = \{1, 2, 3, 4\}, what subset does 1010_2 correspond to? Back: \{1, 3\} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Cloze Representing subsets with bit strings, {0000_2} maps to {\varnothing} \subseteq \{1, 2, 3, 4\}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Cloze Representing subsets with bit strings, {1111_2} maps to {\{1, 2, 3, 4\}} \subseteq \{1, 2, 3, 4\}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

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$$

%%ANKI Basic What does \sum_{k=0}^n (-1)^k \binom{n}{k} evaluate to? Back: 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 does expression \sum_{k=0}^n (-1)^k \binom{n}{k} = 0 indicate about sets? Back: There is an equal number of even-sized subsets as odd-sized subsets. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What isomorphism shows finite sets have an equal number of even-sized subsets as odd-sized subsets? Back: The one-to-one relationship between subsets and bit strings. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given finite set A, how many subsets of even cardinality does A have? Back: Let n = |A|. Then 2^{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 Given finite set A, how many subsets of odd cardinality does A have? Back: Let n = |A|. Then 2^{n-1}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

Stars and Bars

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: ||||*

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 0s as stars and 1s as bars. The above example is equivalently written as: $00100010110$

%%ANKI Basic What symbols are typically used in a stars and bars chart? Back: * and | Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic 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.

END%%

%%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.

END%%

%%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.

END%%

%%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.

END%%

%%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.

END%%

%%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.

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 1, we see every lattice path has a corresponding #Bit Strings.

!lattice-path-example.png

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.

END%%

%%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.

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{14}{7} 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? !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.

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

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 1s and n - k 0s. 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.

END%%

%%ANKI Basic What is a degree-0 polynomial? Back: A constant. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

END%%

%%ANKI Basic What name is given to a degree-0 polynomial? Back: A constant. Reference: Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).

END%%

%%ANKI Basic What name is given to a degree-1 polynomial? Back: A monomial. 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 a degree-2 polynomial? Back: A binomial. 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 1s and n - k 0s. 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%%

%%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.

END%%

References