Chapter 6.1, Problem 51PS

Solution

$\left(a\right)$

To find: The number of combinations of $5$ singers out of original $20$ can be eliminated during the first episode.

  $15,504$

Given information: A televised singing competition picks a winner from $20$ original contestants over the course of five episodes. During each of the first, second, and third episodes, $5$ singers are eliminated by the end of the episode. The fourth episode eliminates more singers, and the winner is selected at the end of the fifth episode.

How many combinations of $5$ singers out of the original $20$ original can be eliminated during the first episode.

Explanation:

The number of combinations of $5$ singers out of the original $20$ can be eliminated during the first episode will be:

  $C{}_5=15,504$

$\left(b\right)$

To find: The number of combinations of $5$ singers out of $15$ singers who started the second episode can be eliminated during the second episode.

  $3003$

Given information: How many combinations of $5$ singers out of $15$ singers who started the second episode can be eliminated during the second episode

Explanation:

The number of combinations of $5$ singers out of $15$ singers who started the second episode can be eliminated during the second episode will be:

  $C{}_5=3,003$

$\left(c\right)$

To find: The number of combinations of singers can be eliminated during the third episode, fourth episode and fifth episode.

  $252,10,3$

Given information: How many combinations of singers can be eliminated during the third episode $?$ During the fourth episode? During the fifth episode $?$

Explanation:

The number of combinations of singers that can be eliminated during the third episode will be:

  $C{}_5=252$

The number of combinations of singers that can be eliminated during the fourth episode will be:

  $C{}_2=10$

The number of combinations of singers that can be eliminated during the fifth episode will be:

  $C{}_2=3$

$\left(d\right)$

To find: The total number of ways in which the $20$ original contestants can be eliminated to produce a winner.

  $351,982,350,700$

Given information: Find the total number of ways in which the $20$ original contestants can be eliminated to produce a winner.

Explanation:

The total number of ways in which the $20$ original contestants can be eliminated to produce a winner will be:

  $C{}_5\times C{}_{15}\times C{}_5\times C{}_2\times C{}_2=351,982,350,700$