Question:
If P(11, r) = P (12, r − 1) find r.
Solution:
$P(11, r)=P(12, r-1)$
$\Rightarrow \frac{11 !}{(11-r) !}=\frac{12 !}{(13-r) !}$
$\Rightarrow \frac{(13-r)}{(11-r) !}=\frac{12 !}{11 !}$
$\Rightarrow \frac{(13-r)(12-r)(11-r) !}{(11-r) !}=\frac{12 \times 11 !}{11 !}$
$\Rightarrow(13-r)(12-r)=12$
$\Rightarrow(13-r)(12-r)=4 \times 3$
On comparing the two sides, we get:
$13-r=4$
$\Rightarrow r=9$
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