Question: The management of a company, a president and four vice-presidents, denoted by the set {P, V1, V2,…

The management of a company, a president and four vice-presidents, denoted by the set {P, V1, V2, V3, V4}, wish to select a committee of 4 people from among themselves. How many ways can this committee be formed? That is, how many 4-person subsets can be formed from a set of 5 people?

How many ways can a 4-person committee be formed using the president and four vice-presidents?

Solution

The problem is about forming a committee from a group of 5 people, which includes a president and four vice-presidents. We need to determine how many ways a 4-person committee can be formed from these 5 people. This is a combination problem. We can use the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. In this problem, \( n = 5 \) and \( r = 4 \). Now, apply the formula: First, plug in the values: \[ C(5, 4) = \frac{5!}{4!(5-4)!} \] Simplify the expression: \[ C(5, 4) = \frac{5!}{4! \cdot 1!} \] Calculate the factorials: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] \[ 1! = 1 \] Substitute back into the expression: \[ C(5, 4) = \frac{120}{24 \cdot 1} \] Simplify further: \[ C(5, 4) = \frac{120}{24} = 5 \] Thus, there are 5 ways to form a 4-person committee from the set of 5 people.