Permutations & Combinations

The value of $5!$ is:

The number of ways to arrange $4$ distinct books on a shelf is:

A menu offers $4$ appetizers, $5$ mains, and $3$ desserts. The number of distinct three-course meals is:

The number of $3$-letter codes that can be formed using letters from $\{A, B, C, D, E\}$ without repetition is:

Determine $n \in \mathbb{N}$, $n \ge 2$, such that $\binom{n}{2} = 15$.

The number of $3$-element subsets of a $5$-element set is:

The sum of all distinct three-digit numbers that can be formed using the digits $\{2, 2, 5\}$ is:

The number of two-digit numbers with distinct digits formable from $\{1, 2, 3, 4, 5\}$ is:

At a meeting, every pair of people shakes hands exactly once. If there are $45$ handshakes total, how many people are there?

Using Pascal's identity $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$, compute $\binom{7}{3} + \binom{7}{4}$.