Question

Evaluate the following binomial coefficients.

$\mathrm{a}.\left(\begin{array}{c}4\\ 1\end{array}\right)\mathrm{b}.\left(\begin{array}{c}6\\ 2\end{array}\right)\mathrm{c}.\left(\begin{array}{c}8\\ 3\end{array}\right)\mathrm{d}.\left(\begin{array}{c}9\\ 6\end{array}\right)$

Short answer

Part (a) 4

Part (b) 15

Part (c) 56

Part (d) 84

Step-by-step solution

Part (a) Step 1. Given information.

The given binomial coefficient is:

$\left(\begin{array}{c}4\\ 1\end{array}\right)$

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

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

Part (b) Step 1. Evaluate the binomial coefficient 62.

$$\begin{gathered} \left(\begin{array}{c}6\\ 2\end{array}\right)=\frac{6!}{2!\times \left(6-2\right)!} \\ =\frac{6!}{2!\times 4!} \\ =\frac{6\times 5}{1\times 2} \\ =15 \end{gathered}$$

Part (c) Step 1. Evaluate the binomial coefficient 83.

$$\begin{gathered} \left(\begin{array}{c}8\\ 3\end{array}\right)=\frac{8!}{3!\times \left(8-3\right)!} \\ =\frac{8!}{3!\times 5!} \\ =\frac{8\times 7\times 6}{1\times 2\times 3} \\ =56 \end{gathered}$$

Part (d) Step 1. Evaluate the binomial coefficient 96.

$$\begin{gathered} \left(\begin{array}{c}9\\ 6\end{array}\right)=\frac{9!}{6!\times \left(9-6\right)!} \\ =\frac{9!}{6!\times 3!} \\ =\frac{9\times 8\times 7}{1\times 2\times 3} \\ =84 \end{gathered}$$