Question

A study group is to be selected from 5 freshmen, 7 sophomores, and 4 juniors. a. If a study group is to consist of 4 freshmen, 6 sophomores, and 2 juniors, how many different ways can the study group be selected? b. If a study group consisting of 6 students is selected, what is the probability that the group will consist of 2 freshmen, 3 sophomores, and 1 junior?

Answer 1

Solution:

Given:

A study group is to be selected from:

5 freshmen,

7 sophomores, and

4 juniors

Total = 16

Part a) how many different ways can the study group be selected if a study group is to consist of:

4 freshmen, 6 sophomores, and 2 juniors

4 freshmen from 5 freshmen can be selected in ₅C₄ ways

6 sophomores from 7 sophomores can be selected in ₇C₆ ways

and

2 juniors from 4 juniors can be selected in ₄C₂ ways

Thus

Total number of Ways = ₅C₄ X ₇C₆ X ₄C₂

Use following combination formula:

Thus

Total number of Ways = ₅C₄ X ₇C₆ X ₄C₂

Total number of Ways =5 X 7 X 6

Total number of Ways = 210

Part b) Find:

the probability that the group will consist of 2 freshmen, 3 sophomores, and 1 junior if a study group consisting of 6 students.

P( 2 freshmen, 3 sophomores, and 1 junior) =.........?

P( 2 freshmen, 3 sophomores, and 1 junior) = [ ₅C₂ X ₇C₃ X ₄C₁ ] / [ ₁₆C₆ ]

and

Thus

P( 2 freshmen, 3 sophomores, and 1 junior) = [ ₅C₂ X ₇C₃ X ₄C₁ ] / [ ₁₆C₆ ]

P( 2 freshmen, 3 sophomores, and 1 junior) = [ 10 X 35 X 4 ] / [ 8008 ]

P( 2 freshmen, 3 sophomores, and 1 junior) = 1400 / 8008

P( 2 freshmen, 3 sophomores, and 1 junior) = 0.174825

P( 2 freshmen, 3 sophomores, and 1 junior) = 0.1748