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

$$\begin{gathered} \Rightarrow \frac{\left(2n-1\right)!}{\left(2n-1-n\right)!}\times \frac{\left(2n+1-n+1\right)!}{\left(2n+1\right)!}=\frac{22}{7} \\ \Rightarrow \frac{\left(2n-1\right)!}{\left(n-1\right)!}\times \frac{\left(n+2\right)!}{\left(2n+1\right)!}=\frac{22}{7} \\ \Rightarrow \frac{\left(2n-1\right)!}{\left(n-1\right)!}\times \frac{\left(n+2\right)\left(n+1\right)\left(n\right)\left(n-1\right)!}{\left(2n+1\right)\left(2n\right)\left(2n-1\right)!}=\frac{22}{7} \\ \Rightarrow \frac{\left(n+2\right)\left(n+1\right)\left(n\right)}{\left(2n+1\right)\left(2n\right)}=\frac{22}{7} \\ \Rightarrow \frac{\left(n+2\right)\left(n+1\right)}{2\left(2n+1\right)}=\frac{22}{7} \\ \Rightarrow 7n^2+21n+14=88n+44 \\ \Rightarrow 7n^2-67n-30=0 \\ \Rightarrow 7n^2-70n+3n-30=0 \\ \Rightarrow \left(n-10\right)\left(7n+3\right)=0 \\ \therefore n=10\mathrm{or}\frac{-3}{7} \\ \text{Since}n\text{cannot be negative, it is equal to10.} \end{gathered}$$