Binomial Coefficient vs Binomial Distribution

The binomial coefficient is the counting piece of the binomial probability formula: it answers how many different arrangements of $k$ successes fit inside $n$ trials. With 5 free throws and 2 makes there are $\binom{5}{2} = \frac{5!}{2!\,3!} = \frac{120}{2(6)} = 10$ orders in which those two makes could happen. The general rule is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ (n choose k). Multiplying that count by $p^k(1-p)^{n-k}$ gives the binomial probability of exactly $k$ successes.

A binomial setting has a fixed number of trials, two outcomes per trial, independent trials, and the same success probability each time. For example, the number of heads in 10 flips of a fair coin is binomial with $n = 10$ trials and success probability $p = 0.5$. The probability of exactly $k$ successes is $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ (n choose k, times p to the power k, times one minus p to the power n minus k). Its mean is $\mu = np$ (n times p).