Question

Evaluate the following binomial coefficients.

$\mathrm{a}.\left(\begin{array}{c}5\\ 3\end{array}\right)\mathrm{b}.\left(\begin{array}{c}10\\ 0\end{array}\right)\mathrm{c}.\left(\begin{array}{c}10\\ 10\end{array}\right)\mathrm{d}.\left(\begin{array}{c}9\\ 5\end{array}\right)$

Short answer

Part (a) 10

Part (b) 1

Part (c) 1

Part (d) 126

Step-by-step solution

Part (a) Step 1. Given information.

The given binomial coefficient is:

$\left(\begin{array}{c}5\\ 3\end{array}\right)$

Part (a) Step 2. Evaluate the binomial coefficient.

$$\begin{gathered} \left(\begin{array}{c}5\\ 3\end{array}\right)=\frac{5!}{3!\times \left(5-3\right)!} \\ =\frac{5!}{3!\times 2!} \\ =\frac{5\times 4}{1\times 2} \\ =10 \end{gathered}$$

Part (b) Step 1. Evaluate the binomial coefficient 100.

$$\begin{gathered} \left(\begin{array}{c}10\\ 0\end{array}\right)=\frac{10!}{0!\times \left(10-0\right)!} \\ =\frac{10!}{0!\times 10!} \\ =\frac{10!}{1\times 10!} \\ =1 \end{gathered}$$

Part (c) Step 1. Evaluate the binomial coefficient 1010.

$$\begin{gathered} \left(\begin{array}{c}10\\ 10\end{array}\right)=\frac{10!}{10!\times \left(10-10\right)!} \\ =\frac{10!}{10!\times 0!} \\ =\frac{10!}{10!\times 1} \\ =1 \end{gathered}$$

Part (d) Step 1. Evaluate the binomial coefficient 95.

$$\begin{gathered} \left(\begin{array}{c}9\\ 5\end{array}\right)=\frac{9!}{5!\times \left(9-5\right)!} \\ =\frac{9!}{5!\times 4!} \\ =\frac{9\times 8\times 7\times 6}{1\times 2\times 3\times 4} \\ =126 \end{gathered}$$