Chapter 11.9, Problem 23E

To determine

To calculate : the number of different groups of students chosen from the class to work on a team project.

Answer to Problem 23E

The number of different groups of students chosen from the class to work on a team project is,

${}_{20}{C}_5=15504$

Explanation of Solution

Given information : A group of 5 students is chosen from a class of 20 students to work on a team project. the possible combination can be,

${}_{20}{C}_5$

Formula used : the number of combination of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$ .

${}_n{C}_r=\frac{{}_n{P}_r}{r!}$

Calculation : the given expression can be simplified as,

$\begin{array}{l}{}_{20}{C}_5=\frac{{}_{20}{P}_5}{5!}\left[\text{use combination formula }\right]\\ {}_{20}{C}_5=\frac{20\cdot 19\cdot 18\cdot 17\cdot 16}{5\cdot 4\cdot 3\cdot 2\cdot 1}\left[{\text{write }}_{\text{20}}{\text{P}}_{\text{5}}\text{ and 5! as products}\right]\\ {}_{20}{C}_5=15504\left[\text{divide out common factor and multiply}\right]\end{array}$

Therefore,the number of different groups of students chosen from the class to work on a team project is,

${}_{20}{C}_5=15504$