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Chapter 11: Problem 45

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 that the first three finishers can come in.

Step by step solution

01

Understanding the Notation

\( _nP_r \) denotes the number of permutations of \( n \) items in groups of \( r \). This is calculated as \( _nP_r = n! / (n-r)! \). Here \( n! \) is the factorial of \( n \) (the product of all positive integers up to \( n \)).
02

Substitute Values

Substitute \( n = 6 \) and \( r = 3 \) into the formula. That gives \( _6P_3 = 6! / (6-3)! \).
03

Calculate Factorials

Calculate the factorials in the formula. \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \), and \( 3! = 3 \times 2 \times 1 = 6. \)
04

Final Calculation

Insert these values back into the formula to get \( _6P_3 = 720 / 6 = 120. \)

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