Question

assume a class has 27 members.
a. in how many ways can a president, a vice president, and a secretary be selected?
b. how many committees of 11 people can be chosen?
a. a president, a vice president, and a secretary can be selected in □ ways.
b. an 11 - person committee can be selected in □ ways.

Explanation:

Part a: Selecting President, VP, Secretary

Step1: Understand permutations

We need to choose 3 distinct positions (president, vice - president, secretary) from 27 members. The number of permutations of \(n\) objects taken \(r\) at a time is given by the formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 27\) and \(r=3\).

Step2: Calculate \(P(27,3)\)

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Part b: Selecting committees of 11 people

Step1: Understand combinations

We need to choose 11 members from 27 for a committee. The number of combinations of \(n\) objects taken \(r\) at a time is given by the formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 27\) and \(r = 11\).

Step2: Calculate \(C(27,11)\)

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Answer:

a. The number of ways to select a president, a vice - president, and a secretary is \(\boldsymbol{17550}\).
b. The number of ways to select an 11 - person committee is \(\boldsymbol{13037895}\).