Problem 57 - College Algebra

Question

Solve by the method of your choice. In a race in which six automobiles are entered and there are no ties, in how many ways can the first four finishers come in?

Step-by-Step Solution

Answer

The first four cars can finish in 360 different ways.

Step 1: Understanding the Problem Statement

The problem is asking for the number of ways the first four cars can finish in a race with six cars, with the stipulation that there can be no ties.

Step 2: Identify the Type of Problem

Because the order in which the cars finish is significant (1st, 2nd, 3rd, and 4th), this problem is a permutation type problem. Permutation is a counting method in combinatorial mathematics where the order of elements is significant.

Step 3: Apply the Formula for Permutations

Use the formula for permutations, which is 'nPr=n!/(n-r)!', where 'n' is the total number of elements, 'r' is the number of elements to choose, and the '!' denotes the factorial operation. In this case, 'n' is 6 (the number of cars) and 'r' is 4 (the number of finishing positions).

Step 4: Calculate the Permutations

Calculate the permutations 6P4 = 6!/(6-4)! = 6*5*4*3/2*1 = 30*12 = 360. So, there are 360 possible ways the first four cars can finish.