Question

If P (15,r-1): P(16,r-2)=3:4, find r.

Answer 1

We have,
P (15,r-1): P(16,r-2)= 3:4
$\Rightarrow \frac{P(15,r-1)}{P(16,r-2)}=\frac{3}{4}\Rightarrow \frac{\frac{15!}{[15-(r-1\left)\right]!}}{\frac{16!}{[16-(r-2\left)\right]!}}=\frac{3}{4}\Rightarrow \frac{\frac{15!}{[16-r]!}}{\frac{16!}{[18-r]!}}=\frac{3}{4}\Rightarrow \frac{15!}{(16-r)!}\times \frac{(18-r)!}{16!}=\frac{3}{4}\Rightarrow \frac{15!(18-r)(17-r)(16-r)!}{(16-r)!\times 16\times 15!}=\frac{3}{4}\Rightarrow \frac{(18-r)(17-r)}{16}\frac{3}{4}\Rightarrow r^2-35r+306=12\Rightarrow r^2-35r+306-12=0\Rightarrow r^2-35r+294=0\Rightarrow r^2-21r-14r+294=0\Rightarrow r(r-21)-14(r-21)=0\Rightarrow (r-21)(r-14)=0\Rightarrow r-14=0[\because r=21\ne 0]\Rightarrow r=14$
Hence, r=14