Question

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

Answer 1

P (15, r − 1):P (16, r − 2) = 3:4

$$\begin{gathered} \Rightarrow \frac{15!}{\left(15-r+1\right)!}\times \frac{(16-r+2)!}{16!}=\frac{3}{4} \\ \Rightarrow \frac{15!}{\left(16-r\right)!}\times \frac{\left(18-r\right)!}{16\times 15!}=\frac{3}{4} \\ \Rightarrow \frac{\left(18-r\right)\left(17-r\right)\left(16-r\right)!}{\left(16-r\right)!\left(16\right)}=\frac{3}{4} \\ \Rightarrow \left(18-r\right)\left(17-r\right)=12 \\ \Rightarrow \left(18-r\right)\left(17-r\right)=4\times 3 \\ \mathrm{On}\mathrm{comparing}\mathrm{the}\mathrm{LHS}\hspace{0.17em}\mathrm{and}\text{the}\mathrm{RHS}\hspace{0.17em}\mathrm{in}\mathrm{above}\mathrm{expression},\mathrm{we}\mathrm{get}: \\ \Rightarrow 18-r=14 \\ \Rightarrow r=14 \end{gathered}$$