Question

Rework problem 35 from the Chapter 2 review exercises in your text, involving auditioning for a play. For this problem, assume 13 males audition, one of them being Karthikey, 7 females audition, one of them being Tiffany, and 5 children audition. The casting director has 4 male roles available, 2 female roles available, and 1 child role available.

(1) How many different ways can these roles be filled from these auditioners?
(2) How many different ways can these roles be filled if exactly one of Karthikey and Tiffany gets a part?
(3) How many different ways can these roles be filled if at least one of Karthikey and Tiffany gets a part?
(4) What is the probability (if the roles are filled at random) of both Karthikey and Tiffany getting a part?

Answer 1

Given,

Number of male persons for audition= 13

Number of female persons for audition= 7

Number of children for audition= 5

ANSWER-1:

There are 4 male roles available, so the casting director has to choose 4 persons out of the 13 male persons available and this can be done in ways.

There are 2 female roles available, so the casting director has to choose 2 persons out of the 7 female persons available and this can be done in ways.

There are 1 child role available, so the casting director has to choose 1 child out of the 5 children available and this can be done in ways.

We have to look at all the possible combinations available for the 7 people selected.

So, the number of different ways these roles can be filled==(13!/(9!4!))(7!/(5!2!))(5!/(4!1!))

=75,075 ways

ANSWER 2:

We have to consider two cases.

Case I: If Karthikey is selected for the role and Tiffany is not selected for the role.

In this case,the number of male persons to be selected for audition reduces to 3 and the number of people from which these 3 persons need to be selected reduces to 12. The 3 persons can be selected in i.e; 220 ways.

If Tiffany is not selected for the role, then number of female persons for audition reduces to 6. So, now we have to choose 2 female persons from the 6 available and this can be done in ,i.e; 15 ways

For children it remains the same.

So, the number of ways these roles can be filled= 220155=16,500 ways.

Case II: If Tiffany gets selected for the role and Karthikey is not selected.

If Karthikey is not selected then the number of male persons for audition reduces to 12. So, now we have to choose 4 male persons from the 12 available and this can be done in ways,i.e; 495 ways.

If Tiffany  gets a part, then number of female persons to be selected for audition reduces to 1 and the number of people from which the 1 female has to be selected reduces to 6. The 1 female can be selected in , i.e; 6 ways.

For children, it remains the same.

So the number of ways, these roles can be filled=49565 ways= 14,850 ways.

So, the number of ways the roles can be filled if exactly one of KARTHIKEY and TIFFANY get a part=14,850+16500 ways

=31350 ways

ANSWER 3:

We have to consider here three cases.

Case I: If Karthikey is selected for the role and Tiffany is not selected for the role.

In this case,the number of male persons to be selected for audition reduces to 3 and the number of people from which these 3 persons need to be selected reduces to 12. The 3 persons can be selected in i.e; 220 ways.

If Tiffany is not selected for the role, then number of female persons for audition reduces to 6. So, now we have to choose 2 female persons from the 6 available and this can be done in ,i.e; 15 ways

For children it remains the same.

So, the number of ways these roles can be filled= 220155=16,500 ways.

Case II: If Tiffany gets selected for the role and Karthikey is not selected.

If Karthikey is not selected then the number of male persons for audition reduces to 12. So, now we have to choose 4 male persons from the 12 available and this can be done in ways,i.e; 495 ways.

If Tiffany  gets a part, then number of female persons to be selected for audition reduces to 1 and the number of people from which the 1 female has to be selected reduces to 6. The 1 female can be selected in , i.e; 6​​​​​​​ ways.

For children, it remains the same.

So the number of ways, these roles can be filled=49565 ways= 14,850 ways.

Case III: If both Tiffany and Karthikey get a part

If Karthikey  gets a part, then number of male persons to be selected for audition reduces to 3 and the number of people from which these 3 persons need to be selected reduces to 12. The 3 persons can be selected in ways.

If Tiffany  gets a part, then number of female persons to be selected for audition reduces to 1 and the number of people from which the 1 female has to be selected reduces to 6. The 1 female can be selected in , i.e; 6 ways.

One child has to be selected from 5 children and that can be done in ,i.e; 5 ways.

So, the total number of ways in which the roles can be filled if both  karthikey and Tiffany gets

selected=56 ways

=(12!(9!3!))30 ways

= 6600 ways

So, the number of different ways the roles can be filled if atleast one of Tiffany and Karthikey get a part=16,500+14,850+6600

=37950 ways

ANSWER 4:

Probability of both karthikey and Tiffany getting a part=

  

No. of different ways the roles can be filled ,i.e; 75075

= 8/91