Question

Evaluate the following binomial coefficients.

$\mathrm{a}.\left(\begin{array}{c}5\\ 2\end{array}\right)\mathrm{b}.\left(\begin{array}{c}7\\ 4\end{array}\right)\mathrm{c}.\left(\begin{array}{c}10\\ 3\end{array}\right)\mathrm{d}.\left(\begin{array}{c}12\\ 5\end{array}\right)$

Short answer

Part (a) 10

Part (b) 35

Part (c) 120

Part (d) 792

Step-by-step solution

Part (a) Step 1. Given information.

The given binomial coefficient is:

$\left(\begin{array}{c}5\\ 2\end{array}\right)$

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

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

Part (b) Step 1. Evaluate the binomial coefficient 74.

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

Part (c) Step 1. Evaluate the binomial coefficient 103.

$$\begin{gathered} \left(\begin{array}{c}10\\ 3\end{array}\right)=\frac{10!}{3!\times \left(10-3\right)!} \\ =\frac{10!}{3!\times 7!} \\ =\frac{10\times 9\times 8}{1\times 2\times 3} \\ =120 \end{gathered}$$

Part (d) Step 1. Evaluate the binomial coefficient 125.

$$\begin{gathered} \left(\begin{array}{c}12\\ 5\end{array}\right)=\frac{12!}{5!\times \left(12-5\right)!} \\ =\frac{12!}{5!\times 7!} \\ =\frac{12\times 11\times 10\times 9\times 8}{1\times 2\times 3\times 4\times 5} \\ =792 \end{gathered}$$