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Chapter 11: Problem 62

Write a word problem that can be solved by evaluating \({ }_{7} P_{3}\).

Short Answer

Expert verified
There are 210 possible ways to award the medals among the 7 runners.

Step by step solution

01

Problem Statement

In a 100-meter race, there are 7 participants and the first three positions will be awarded gold, silver, and bronze. In how many ways can these 3 medals be awarded between the 7 runners?
02

Applying Permutation

One way of solving this problem is by calculating permutation. The permutation formula (assuming no repetition) is given as: \[ nPr = n! / (n - r)!\] where n is the total number of items to choose from, r is the number of items to choose, '!' denotes 'factorial', and nPr denotes permutation. Here, there are 7 runners and 3 positions to be filled, so \[{}_{7}P_{3} = 7! / (7-3)!\].
03

Calculating Factorial

Now calculate the factorial values. Factorial essentially means multiplying all positive integers from 1 through that number. So, 7! equals to \[7*6*5*4*3*2*1 = 5040\] and 4! equals to \[4*3*2*1 = 24\].
04

Final Calculation

Substitute the calculated factorial values into the permutation equation from Step 2, which gives: \[{}_{7}P_{3} = 5040 / 24 = 210\].

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