The binomial coefficient calculates combinations using the formula n!/(x!(n-x)!), where n represents the total number of items, and x represents the number of items to choose. In the example given, 5 over 2 and 5 over 3 both equal 10, which illustrates that there are 10 ways to choose 2 or 3 items from a set of 5. The confusion initially expressed was resolved, indicating a better understanding of the concept. This formula is fundamental in combinatorics for determining the number of ways to select items without regard to order. Understanding this calculation is essential for solving problems related to combinations.