Assume that a committee consists of 12 people including the chairperson and that the chairperson must select a committee of 3 people from the other 11 members of the committee.
In how many ways can this be done?
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To determine the number of ways to select a committee of 3 people from the remaining 11 members, we can use the concept of combinations.
The formula to calculate the number of combinations is:
C(n, r) = n! / (r!(n - r)!)
Where:
- n is the total number of elements
- r is the number of elements to be chosen
In this case, we have 11 members from which we need to select a committee of 3 people. So, using the formula above, we have:
C(11, 3) = 11! / (3!(11 - 3)!)
Simplifying the expression further:
C(11, 3) = 11! / (3!8!)
Now, let's expand the factorials:
11! = 11 × 10 × 9 × 8!
3! = 3 × 2 × 1
Plugging these values into the formula:
C(11, 3) = (11 × 10 × 9 × 8!) / (3 × 2 × 1 × 8!)
Simplifying the expression:
C(11, 3) = (11 × 10 × 9) / (3 × 2 × 1)
Doing the arithmetic:
C(11, 3) = 165
Therefore, there are 165 different ways to select a committee of 3 people from the remaining 11 members.