# In order to conduct an experiment, five students are randomly selected from a class of 20. How many different groups of five students are possible?

Jan 15, 2018

$15504$

#### Explanation:

This can be done using the choose function.

The number of combinations are given by:

([n],[k])=(n!)/(k!(n-k)!)

where $n$ is the total number of students and $k$ is the number of students to be picked. So we have $n = 20$ and $k = 5$:

(,)=(20!)/(5!(20-5)!)=(20!)/(5!15!)

Evaluate directly with a calculator:

(20!)/(5!15!)=15504

we can simplify this before calculation by hand:

(20!)/(5!15!)=(20times19times...times2times1)/(5times4times3times2times1times(15times...times1)

$= \frac{\left(20 \times \ldots \times 16\right) \left(15 \times \ldots \times 1\right)}{\left(5 \times \ldots \times 1\right) \left(15 \times \ldots \times 1\right)} = \frac{\left(20 \times \ldots \times 16\right) \cancel{15 \times \ldots \times 1}}{\left(5 \times \ldots \times 1\right) \cancel{15 \times \ldots \times 1}}$

$= \frac{\left(\textcolor{red}{20} \times 19 \times \textcolor{b l u e}{18} \times 17 \times \textcolor{g r e e n}{16}\right)}{\left(\textcolor{red}{5} \times \textcolor{g r e e n}{4} \times 3 \times \textcolor{b l u e}{2} \times 1\right)}$

Simplify the numbers matched up by color:

$= \frac{\left(4 \times 19 \times \textcolor{g r e e n}{9} \times 17 \times 4\right)}{\left(1 \times 1 \times \textcolor{g r e e n}{3} \times 1 \times 1\right)}$

$= \frac{\left(4 \times 19 \times 3 \times 17 \times 4\right)}{\left(1 \times 1 \times 1 \times 1 \times 1\right)}$

$= 3 \times 4 \times 4 \times 17 \times 19$

$= 48 \times 323 = 15504$