26 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 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 of a given size? 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.
!
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?
!
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%%
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%%
Bibliography
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.