Problem 90 - Precalculus with Limits

Question

Without calculating the numbers, determine which of the following is greater. Explain. (a) The number of combinations of \( 10 \) elements taken six at a time (b) The number of permutations of \( 10 \) elements taken six at a time

Step-by-Step Solution

Answer

The number of permutations of 10 elements taken six at a time is greater than the number of combinations.

Step 1: Understanding Combinations and Permutations

In combinatorics, combinations and permutations are two different ways of selecting items from a set. A combination is a selection where order does not matter, meaning each selection of items is unique and the order in which the items are selected does not matter. A permutation, on the other hand, is a selection where order does matter. For every unique set of items, there will always be more ways to arrange (permutations) them than there are to simply select (combinations) them.

Step 2: Determining Which is Greater

Since a permutation takes into account the order of items and a combination does not, for a given set of items there will always be more permutations than combinations. This is because for every combination, there are multiple ways those items can be ordered, each of which counts as a distinct permutation.