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:

$n C_{r}=\frac{n !}{(n-r) ! r !}$

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

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

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

$n !=n \times(n-1) \times(n-2) \dots \times 2 \times 1$

Therefore,

$20 C_{3}=\frac{20 \times 19 \times 18 \times 17 \ldots \ldots \ldots 2 \times 1}{17 \times 16 \times 15 \ldots .2 \times 1 \times 3 !}$

$20 C_{3}=\frac{20 \times 19 \times 18}{3 !}=\frac{20 \times 19 \times 18}{3 \times 2 \times 1}=1140$

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