Concept explainers
Selecting a Committee There are 7 women and 5 men in a department. How many ways can a committee of 4 people be selected? How many ways can this committee be selected if there must be 2 men and 2 women on the committee? How many ways can this committee be selected if there must be at least 2 women on the committee?
The number of ways to select 4 people for the committee, 2 men and 2 women for the committee and at least 2 women for the committee.
Answer to Problem 43E
There are 495 ways to select 4 people out of 12 people.
There are 210 ways to select 2 men and 2 women
There are 420 ways to select at least 2 women.
Explanation of Solution
Given info:
A committee must be formed with 4 members. There are 7 women and 5 men available for the selection.
Calculation:
Combinations:
A combination is an arrangement of n objects in r ways where the order of arrangement is not considered.
Where, n is the total number of objects and r is number of ways in which n objects can be selected.
The number of ways to select 4 people out of 12 people:
Substitute n as 12 and r as 4.
Thus, there are 495 ways to select 4 people out of 12 people.
The number of ways to select 2 women and 2 men out of 7 women and 5 men:
Substitute n as 7 and r as 2.
Thus, there are 21 ways to select 2 women out of 7 women.
Substitute n as 5 and r as 2.
Thus, there are 10 ways to select 2 men out of 5 men.
Fundamental counting rule:
The number of ways in which a sequence of n events occur if the first event can occur in
The number ways to select 2 men and 2 women is given below:
Thus, there are 210 ways to select 2 men and 2 women.
The number of ways to select at least 2 women for the committee:
There can be 2 women, 3 women and 4 women in the committee.
The number of ways to select 2 women and 2 men is 210 ways.
The number of ways to select 3 women and 1 man
Substitute n as 7 and r as 3.
Thus, there are 35 ways to select 3 women out of 7 women.
Substitute n as 5 and r as 1.
The number ways to select 2 men and 2 women is given below:
Thus, there are 175 ways to select 3 women and 1 man.
The number of ways to select 4 women and 0 men
Substitute n as 7 and r as 4.
Thus, there are 35 ways to select 4 women out of 7 women.
The number of ways to select at least 2 women for the committee is given below:
Thus, there are 420 ways to select at least 2 women for the committee.
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Chapter 4 Solutions
Elementary Statistics: A Step-by-Step Approach with Formula Card
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