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

– There are $25$ members in a club.

– In how many ways can $4$ members be chosen to serve in an executive committee?

– In how many ways can a president, vice president, secretary, and treasurer of the club be chosen so that each person can only hold a single office at a time?

The aim of this question is to find the number of ways for which an executive committee can be served by $4$ members.

For the other part, we have to find a number of ways to choose a president, vice president, etc without giving the same position to $2$ members

In order to correctly solve this problem, we need to understand the concept of Permutation and Combination.

A combination in mathematics is the arrangement of its given members irrespective of their order.

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$

$C\left(n,r\right)$ = Number of combinations

$n$ = Total number of objects

$r$ = Selected object

A permutation in mathematics is the arrangement of its members in a definite order. Here, the order of the members matters and is arranged in a linear manner. It is also called an Ordered Combination, and the difference between the two is in order.

For example, the PIN of your mobile is $6215$ and if you enter $5216$ it won’t unlock as it is a different ordering (permutation).

$nP_r\\=\frac{n!}{\left(n-r\right)!}$

$n$ = Total number of objects

$r$ = Selected object

$nP_r$ = Permutation

$(a)$ Find the number of ways for which an executive committee can be served by $4$ members. Here, as the order of members does not matter, we will use combination formula.

$n=25$

The committee should be of $4$ members, $r=4$

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$

Putting values of $n$ and $r$ here, we get:

$C\left(25,4\right)=\frac{25!}{4!\left(25-4\right)!}$

$C\left(25,4\right)=\frac{25!}{4!21!}$

$C\left(25,4\right)=12,650$

The number of ways to select the committee of $4$ members $=12,650$

$(b)$ To find out the number of ways to select the club members for a president, vice president, secretary, and treasurer of the club, the order of members is significant, so we will use the definition of permutation.

Total number of club members $=n=25$

Designated positions for which members are to be selected $=r=4$

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Putting values of $n$ and $r$:

$P\left(25,4\right)=\frac{25!}{\left(25-4\right)!}$

$P\left(25,4\right)=\frac{25!}{21!}$

$P\left(25,5\right)=\frac{25 \times 24 \times 23 \times 22 \times 21!}{21!}$

$P\left(25,5\right)=25 \times 24 \times 23 \times 22$

$P\left(25,5\right)=303,600$

The number of ways to select the club members for a president, vice president, secretary, and treasurer of the club $=303,600$.

Numerical Results

The number of ways to choose $4$ members of the club to serve on an executive committee is $12,650$

The number of ways to select the club members for a president, vice president, secretary, and treasurer so that no person can hold more than one office is $303,600$.

Example

A group of $3$ athletes is $P$, $Q$, $R$. In how many ways can a team of $2$ members be formed?

Here, as the order of members is not important, we will use the Combination formula.

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$

Putting values of $n$ and $r$:

$n=3$

$r=2$

$C\left(3,2 \right)=\frac{3!}{2!\left(3-2\right)!}$

$C\left(3,2 \right)=3$