Question

Evaluate the following binomial coefficients.

$\mathrm{a}.\left(\begin{array}{c}3\\ 2\end{array}\right)\mathrm{b}.\left(\begin{array}{c}6\\ 0\end{array}\right)\mathrm{c}.\left(\begin{array}{c}6\\ 6\end{array}\right)\mathrm{d}.\left(\begin{array}{c}7\\ 3\end{array}\right)$

Short answer

Part (a) 3

Part (b) 1

Part (c) 1

Part (d) 35

Step-by-step solution

Part (a) Step 1. Given information.

The given binomial coefficient is:

$\left(\begin{array}{c}3\\ 2\end{array}\right)$

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

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

Part (b) Step 1. Evaluate the binomial coefficient 60.

$$\begin{gathered} \left(\begin{array}{c}6\\ 0\end{array}\right)=\frac{6!}{0!\times \left(6-0\right)!} \\ =\frac{6!}{0!\times 6!} \\ =1 \end{gathered}$$

Part (c) Step 1. Evaluate the binomial coefficient 66.

$$\begin{gathered} \left(\begin{array}{c}6\\ 6\end{array}\right)=\frac{6!}{6!\times \left(6-6\right)!} \\ =\frac{6!}{6!\times 0!} \\ =1 \end{gathered}$$

Part (d) Step 1. Evaluate the binomial coefficient 73.

$$\begin{gathered} \left(\begin{array}{c}7\\ 3\end{array}\right)=\frac{7!}{3!\times \left(7-3\right)!} \\ =\frac{7!}{3!\times 4!} \\ =\frac{7\times 6\times 5}{1\times 2\times 3} \\ =35 \end{gathered}$$