Chapter 9.1, Problem 46E

Solution

To identify: The number of different ways to choose fifth through first places from 10 Miss America finalists.

The options are:

  $\left(a\right)50$

  $\left(b\right)120$

  $\left(c\right)252$

  $\left(d\right)30,240$

  $\left(e\right)3,628,800$

The correct option is $\left(d\right)30,240$ .

Given information:

There are 10 Miss America finalists from which 5 places are to be filled.

Explanation:

Total number of finalists: 10

The first place can be filled with any of the 10 finalists. Then the number of ways for filling the first place is: 10

The second place can be filled with any of the remaining 9 finalists. Then the number of ways for filling the second place is: 9

The third place can be filled with any of the remaining 8 finalists. Then the number of ways for filling the third place is: 8

The fourth place can be filled with any of the remaining 7 finalists. Then the number of ways for filling the fourth place is: 7

The fifth place can be filled with any of the remaining 6 finalists. Then the number of ways for filling the fifth place is: 6

Then the total number of ways will become: $10\cdot 9\cdot 8\cdot 7\cdot 6=30,240$

Therefore, there are 30,240 ways in which five places can be filled. Option $\left(d\right)30,240$ should be the correct answer.