Question

b) ( _np_5 = 13_np_4 )

Explanation:

Step1: Recall permutation formula

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

Step2: Substitute k=5 and k=4

$$\frac{n!}{(n-5)!} = 13 \times \frac{n!}{(n-4)!}$$

Step3: Simplify using $(n-4)!=(n-4)(n-5)!$

$$\frac{n!}{(n-5)!} = 13 \times \frac{n!}{(n-4)(n-5)!}$$
Cancel $\frac{n!}{(n-5)!}$ from both sides:
$$1 = \frac{13}{n-4}$$

Step4: Solve for n

Multiply both sides by $(n-4)$:
$$n-4 = 13$$
$$n = 13 + 4$$

Answer:

$n=17$