Question

A club of twenty students wants to pick a three person subcommittee. How many ways can this be done?

Answer 1

1140 ways are there to select three person subcommittee from a club of twenty students

Solution:

Given that a club of twenty students wants to pick a three person subcommittee

To find: number of ways this can be done

We have to use combinations formula to solve the given sum

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected

The formula to calculate combinations is:

`n C_(r)=(n !)/((n-r) ! r !)`

where n represents the number of items, and r represents the number of items being chosen at a time

Here we have to choose 3 persons from 20 students

So, n = 20 and r = 3

`\begin{aligned}&20 C_(3)=(20 !)/((20-3) ! 3 !)\\\\&20 C_(3)=(20 !)/(17 ! 3 !)\end{aligned}`

To get the factorial of a number n ,the given formula is used,

`n !=n *(n-1) *(n-2) \dots * 2 * 1`

Therefore,

`20 C_(3)=(20 * 19 * 18 * 17 \ldots \ldots \ldots 2 * 1)/(17 * 16 * 15 \ldots .2 * 1 * 3 !)`

`20 C_(3)=(20 * 19 * 18)/(3 !)=(20 * 19 * 18)/(3 * 2 * 1)=1140`

Thus 1140 ways are there to select three person subcommittee from a club of twenty students