Problem 45 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) In a race in which six automobiles are entered and there are no ties, in how many ways can the first three finishers come in?

Step-by-Step Solution

Answer

There are 120 different ways the first three finishers can come in.

Step 1: Identify the total number of items \(n\)

There are 6 automobiles entered into the race. So, the total number of items or automobiles, \(n\), is 6.

Step 2: Identify the number of items to choose from \(r\)

The exercise asks for the number of ways the first three finishers can come in. Therefore, the number of items to choose from, \(r\), is 3.

Step 3: Use permutation formula \(_{n} P_{r}\)

Substitute the values of \(n\) and \(r\) into the formula. The formula becomes \(_{6} P_{3} = \frac{6!}{(6-3)!}\)

Step 4: Perform Factorial Calculation

First, calculate the factorial of 6, which is \(6! = 6*5*4*3*2*1 = 720\) and then of 3, which is \(3! = 3*2*1 = 6\). Then divide \(720\) by \(6\) to get the answer.

Step 5: Apply the division

The division becomes \(720 / 6 = 120\). So, there are 120 ways that the first three finishers can come in.