Question

A club has \({\bf{25}}\) members.

a) How many ways are there to choose four members of the club to serve on an executive committee?

b) How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?

Short answer

a) The possible number of ways is \(12,650\).

b) The possible number of ways is \(303,600\).

Step-by-step solution

Given data

Number of members in club \( = 25\)

Concept of Combination

A combination is a selection of items from a set that has distinct members.

Formula:

\(_n{C_r} = \frac{{n!}}{{r!(n - r)!}}\)

Calculation to choose executive committee

a)

Number of members in club \( = 25\)

Find the number of ways to choose \(4\) members, use the formula of combination:

Number of ways \( = C(25,4)\)

\(\begin{array}{l}C(25,4) = \frac{{25!}}{{4!(25 - 4)!}}\\C(25,4) = \frac{{25!}}{{4!21!}}\\C(25,4) = 12,650\end{array}\)

Hence, the possible number of ways is \(12,650\).

Calculation to choose a president, vice president, secretary, and treasurer of the club

b)

The formula of permutation is\(\;P(n,r) = \frac{{n!}}{{(n - r)!}}\).

Number of members in club \( = 25\).

Find the number of ways to choose \(4\) members (president, vice president, secretary and treasurer of the club), use the formula of permutation:

Number of ways \( = P(25,4)\)

\(\begin{array}{l}P(25,4) = \frac{{25!}}{{(25 - 4)!}}\\P(25,4) = \frac{{25!}}{{21!}}\\P(25,4) = 303,600\end{array}\)

Hence, the possible number of ways is \(303,600\).