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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
\[\Rightarrow \frac{15!}{\left( 15 - r + 1 \right)!} \times \frac{(16 - r + 2)!}{16!} = \frac{3}{4}\]
\[ \Rightarrow \frac{15!}{\left( 16 - r \right)!} \times \frac{\left( 18 - r \right)!}{16 \times 15!} = \frac{3}{4}\]
\[ \Rightarrow \frac{\left( 18 - r \right)\left( 17 - r \right)\left( 16 - r \right)!}{\left( 16 - r \right)!\left( 16 \right)} = \frac{3}{4}\]
\[ \Rightarrow \left( 18 - r \right)\left( 17 - r \right) = 12\]
\[ \Rightarrow \left( 18 - r \right)\left( 17 - r \right) = 4 \times 3\]
\[\text{On comparing the LHS and the RHS in above expression, we get}: \]
\[ \Rightarrow 18 - r = 14\]
\[ \Rightarrow r = 14\]
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