Question

If P (2n − 1, n) : P (2n + 1, n − 1) = 22 : 7 find n.

Answer 1

P (2n − 1, n):P (2n + 1, n − 1) = 22:7

$$\Rightarrow \frac{(2n-1)!}{(2n-1-n)!}\times \frac{(2n+1-n+1)!}{(2n+1)!}=\frac{22}{7}$$
$$\Rightarrow \frac{(2n-1)!}{(n-1)!}\times \frac{(n+2)!}{(2n+1)!}=\frac{22}{7}$$
$$\Rightarrow \frac{(2n-1)!}{(n-1)!}\times \frac{(n+2)(n+1)\left(n\right)(n-1)!}{(2n+1)\left(2n\right)(2n-1)!}=\frac{22}{7}$$
$$\Rightarrow \frac{(n+2)(n+1)\left(n\right)}{(2n+1)\left(2n\right)}=\frac{22}{7}$$
$$\Rightarrow \frac{(n+2)(n+1)}{2(2n+1)}=\frac{22}{7}$$
$$\Rightarrow 7n^2+21n+14=88n+44$$
$$\Rightarrow 7n^2-67n-30=0$$
$$\Rightarrow 7n^2-70n+3n-30=0$$
$$\Rightarrow (n-10)(7n+3)=0$$
$$\therefore n=10or\frac{-3}{7}$$
$$\text{Sincencannot be negative, it is equal to10}.$$