Question

evaluate \\( _{12}c_4 \\) and \\( _{10}p_4 \\). (if necessary, consult a list of formulas.) \\( _{12}c_4 = \square \\) \\( _{10}p_4 = \square \\)

Explanation:

Step1: Recall combination formula

The combination formula is $_{n}\text{C}_{k} = \frac{n!}{k!(n-k)!}$

Step2: Calculate $_{12}\text{C}_{4}$

$$\begin{align*} _{12}\text{C}_{4}&=\frac{12!}{4!(12-4)!}\\ &=\frac{12!}{4!8!}\\ &=\frac{12\times11\times10\times9\times8!}{4\times3\times2\times1\times8!}\\ &=\frac{12\times11\times10\times9}{4\times3\times2\times1}\\ &=495 \end{align*}$$

Step3: Recall permutation formula

The permutation formula is $_{n}\text{P}_{k} = \frac{n!}{(n-k)!}$

Step4: Calculate $_{10}\text{P}_{4}$

$$\begin{align*} _{10}\text{P}_{4}&=\frac{10!}{(10-4)!}\\ &=\frac{10!}{6!}\\ &=10\times9\times8\times7\\ &=5040 \end{align*}$$

Answer:

$_{12}\text{C}_{4}=495$
$_{10}\text{P}_{4}=5040$