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Chapter 8: Problem 45

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?

Short Answer

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

Step by step solution

01

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.
02

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.
03

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

Substitute the values of \(n\) and \(r\) into the formula. The formula becomes \(_{6} P_{3} = \frac{6!}{(6-3)!}\)
04

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.
05

Apply the division

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

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Most popular questions from this chapter

Exercises will help you prepare for the material covered in the next section. In Exercises \(112-113,\) show that $$ 1+2+3+\cdots+n-\frac{n(n+1)}{2} $$is true for the given value of \(n .\) $$n-3: \text { Show that } 1+2+3-\frac{3(3+1)}{2}.$$

Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\) Hint: Write \(x^{2}+x+1\) as \(x^{2}+(x+1)\).

Exercises 86-88 will help you prepare for the material covered in the next section. $$\text { Evaluate } \frac{n !}{(n-r) !} \text { for } n-20 \text { and } r-3$$.

Use the formula for \(_{n} C_{r}\) to solve A four-person committee is to be elected from an organization's membership of 11 people. How many different committees are possible?

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