If 5 P(4, n) = 6. P (5, n − 1), find n.

5 P(4, n) = 6. P (5, n − 1)

5 4Pn = 65Pn-">−-1

$\Rightarrow 5 \times \frac{4 !}{(4-n) !}=6 \times \frac{5 !}{(5-n+1) !}$

$\Rightarrow 5 \times \frac{(6-n) !}{(4-n) !}=6 \times \frac{5 !}{4 !}$

$\Rightarrow 5 \times \frac{(6-n)(6-n-1)(6-n-2) !}{(4-n)}=6 \times \frac{5 \times 4 !}{4 !}$

$\Rightarrow 5 \times \frac{(6-n)(5-n)(4-n) !}{(4-n)}=6 \times 5$

$\Rightarrow(6-n)(5-n)=6$

$\Rightarrow(6-n)(5-n)=3 \times 2$

On comparing the LHS and the RHS, we get:

$\Rightarrow 6-n=3$

$\Rightarrow n=3$