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

P (15, r − 1):P (16, r − 2) = 3:4

$\Rightarrow \frac{15 !}{(15-r+1) !} \times \frac{(16-r+2) !}{16 !}=\frac{3}{4}$

$\Rightarrow \frac{15 !}{(16-r) !} \times \frac{(18-r) !}{16 \times 15 !}=\frac{3}{4}$

$\Rightarrow \frac{(18-r)(17-r)(16-r) !}{(16-r) !(16)}=\frac{3}{4}$

$\Rightarrow(18-r)(17-r)=12$

$\Rightarrow(18-r)(17-r)=4 \times 3$

On comparing the LHS and the RHS in above expression, we get:

$\Rightarrow 18-r=14$

$\Rightarrow r=14$