How do binomial coefficients relate to the probability of repeated independent events?

In combinatorics, binomial coefficients represent the number of ways to choose a subset of $k$ objects from a set of $n$ objects without regard to order, denoted as ${}_n{C}_k$ or "n choose k".1

In the context of probability, these coefficients are fundamental to calculating the likelihood of outcomes in repeated independent events, such as those described by the binomial distribution. When performing $n$ independent trials where each trial has a probability $p$ of success and $q=1-p$ of failure, the binomial coefficient ${}_n{C}_k$ counts the number of distinct sequences or "orders" in which exactly $k$ successes can occur.23

The probability of achieving exactly $k$ successes in $n$ trials is given by the formula:
$$P(X=k){=}_n{C}_k·p^k·q^{n-k}$$
This expression corresponds to the $k$th term in the expansion of the binomial theorem expression $(p+q{)}^n$. Essentially, the binomial coefficient acts as a multiplier that accounts for all possible favorable combinations of successes and failures within the total number of possible outcomes.132