Discrete Math: Combinatorics

Evaluate the binomial coefficient $\binom{8}{3}$.

Evaluate the binomial coefficient $\binom{4}{0}$.

Evaluate the binomial coefficient inom{7}{5}.

Evaluate the binomial coefficient when $n$ and $r$ are the same, for example, $\binom{9}{9}$.

Find the coefficient of the $x^4y^{12}$ term in the expansion of $(5x^2-2y^3)^6$.

Determine the coefficient of the third term in the binomial expansion of $(2x + y)^5$.

Expand $(x + 2)^4$ using Pascal's triangle.

Expand $(2x-3y)^5$ using Pascal's triangle.

Prove the recursive identity for binomial coefficients: inom{n}{k} = inom{n-1}{k} + inom{n-1}{k-1} given the restrictions $n extgreater 0$ and $0 extless k extless n$.

Prove that ${5 \choose 3} \times {3 \choose 2} = {5 \choose 2} \times {3 \choose 1}$.

Prove that the sum of binomial coefficients for a set of size $n$ equals $2^n$, i.e., sum_{k=0}^{n} {n \choose k} = 2^n.

How many different outfits can Mike have with two pants, three shirts, and two pairs of boots?

Given a set A containing numbers 1 through 6, select three objects in an ordered sequence without repetition.

How many ways can this be done?

From a set A containing numbers 1 through 6, determine how many unordered sequences of three objects can be selected without repetition.

From a set A containing numbers 1 through 6, determine how many sequences of three objects can be selected with repetition and unordered.

From a set A with numbers 1 through 6, how many strictly decreasing sequences of three objects can be formed?

Consider the word formed by the letters in 'memory'. How many unique arrangements of these letters are possible?

Consider the word 'memory'. How many arrangements contain the word 'I'?

Given 12 books, how can they be distributed among 4 children?

Last year, 26 members of a certain club traveled to England, 26 members traveled to France, and 32 members traveled to Italy. No members traveled to both England and France, 6 members traveled to both England and Italy, and 11 members traveled to both France and Italy. How many members of the club traveled to at least one of these three countries last year?