Question:
If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
Solution:
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:
$\Rightarrow 18-r=14$
$\Rightarrow r=14$
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