This post will give an intuitive interpretation of the presence of the combination formula (which equals the binomial coefficient) in math problem solutions and probability distributions that are seemingly unrelated to combinations. For example, the binomial coefficient shows up in the probability mass function of the binomial distribution and the negative binomial distribution.

Wikipedia has the following definition for *combination*:

In mathematics a

combinationis a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruit, an apple, orange and pear say, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally ak–combinationof a setSis a subset ofkdistinct elements ofS. If the set hasnelements the number ofk-combinations is equal to the binomial coefficient

The binomial coefficient indexed by n and k is denoted . The formula below, for evaluating binomial coefficients, uses factorials.