Question

Suppose you have n items from which you choose r at a time. Explain why you must divide the number of permutations n!(n−r)! by r! to find the number of combinations.

A. To account for repeated selections
B. To account for order of selection
C. To account for the number of items
D. To account for the size of the group

Answer 1

To find the number of combinations, we divide the number of permutations by r! to correct for the overcounting caused by different orderings of the same items, since for combinations, order doesn't matter.

Step-by-step explanation:

When choosing r items from a set of n without considering the order, we find the number of combinations. To arrive at this number, we start with the number of permutations, which is n! divided by (n-r)!, because this gives us all the possible ways we can arrange r items out of n. However, since in combinations the order doesn't matter, we divide further by r! to correct for the fact that we've counted each unique group of r items r! times—once for each possible order.

This division adjusts for the overcounting inherent in permutations, where different orders of the same items are counted as distinct. By dividing by r!, we effectively disregard the sequence, leaving us with a count of unique selections. This aligns with the concept of combinations, which are selections where the order of the elements is irrelevant.