Graduate Management Admission Test (GMAT) Practice Test

What is the role of factorials in calculating combinations when grouping people?

1. To simplify calculations

2. To create unique arrangements

3. To avoid double counting

4. To determine probability

The correct answer is: To avoid double counting

Factorials play a crucial role in calculating combinations, particularly in ensuring that the calculations accurately reflect unique groupings of individuals without redundancy. When computing combinations, the primary goal is to select a subset of items from a larger set without regard to the order of selection. This is essential because different arrangements of the same group of people should not be counted multiple times; for example, selecting person A and person B is the same as selecting person B and person A. The formula for combinations utilizes factorials in the following way: for a set of n people, the number of ways to choose r individuals is calculated using the formula \( C(n, r) = \frac{n!}{r!(n-r)!} \). In this formula, the factorial of n (n!) represents all possible arrangements of n people, while r! accounts for the arrangements of the r chosen individuals, and (n-r)! accounts for the arrangements of the people not chosen. By dividing by these additional factorials, the formula effectively eliminates duplicate counts that arise from different orders of the same combination. The nature of combinations necessitates that the selection must be distinct to provide true insight into groupings, and it is through this function of factorials that mathematicians can accurately derive