Chapter 9, Problem 19CLT

a.

To determine

To find the binomial coefficient $C{}_{18}$ .

Answer to Problem 19CLT

  $190$

Explanation of Solution

Given: The binomial coefficient $C{}_{18}$ .

Formula used:

Binomial coefficient formula, $C{}_r=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Use the binomial coefficient formula to find the value of $C{}_{18}$ as follows:

  $\begin{array}{c}C{}_{18}=\frac{20!}{18!\left(20-18\right)!}\\ =\frac{20!}{18!\cdot 2!}\\ =\frac{20\cdot 19\cdot 18!}{18!\cdot 2!}\\ =\frac{20\cdot 19}{2!}\\ =\frac{20\cdot 19}{2\cdot 1}\\ =10\cdot 19\\ =190\end{array}$

b.

To determine

To find the binomial coefficient $\left(\begin{array}{c}20\\ 2\end{array}\right)$ .

Answer to Problem 19CLT

  $190$

Explanation of Solution

Given: The binomial coefficient $\left(\begin{array}{c}20\\ 2\end{array}\right)$ .

Formula used:

Binomial coefficient formula, $\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Use the binomial coefficient formula to find the value of $\left(\begin{array}{c}20\\ 2\end{array}\right)$ as follows:

  $\begin{array}{c}\left(\begin{array}{c}20\\ 2\end{array}\right)=\frac{20!}{2!\left(20-2\right)!}\\ =\frac{20!}{2!\cdot 18!}\\ =\frac{20\cdot 19\cdot 18!}{2!\cdot 18!}\\ =\frac{20\cdot 19}{2!}\\ =\frac{20\cdot 19}{2\cdot 1}\\ =10\cdot 19\\ =190\end{array}$