Problem 52 - Algebra and Trigonometry

Question

There are 14 standbys who hope to get seats on a flight, but only 6 seats are available on the plane. How many different ways can the 6 people be selected?

Step-by-Step Solution

Answer

The number of ways to select 6 people from a group of 14 is 3003.

Step 1: Understanding the Problem

The exercise is about selecting 6 people out of 14, this is a problem of combinations in statistics. The number of ways to select r objects from a total of n objects is given by the combination formula \(^nC_r = \frac{n!}{r!(n - r)!}\). In this exercise, n = 14 and r = 6.

Step 2: Applying the Combination Formula

Apply the combination formula to the given problem. Substitute n = 14 and r = 6 in the formula \(^nC_r = \frac{n!}{r!(n - r)!}\). So, the expression becomes \(^{14}C_6 = \frac{14!}{6!(14 - 6)!}\).

Step 3: Calculate the Factorials

Next, calculate the factorials i.e., 14!, 6! and (14-6)!. Remember that the factorial of a non-negative integer n, denoted by n!, is essentially the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1.

Step 4: Compute the Expression

Once you have calculated the factorials, divide the factorial of 14 by the product of the factorial of 6 and the factorial of (14 - 6). This will give the total number of combinations possible to select 6 people from a group of 14.