Question

If P(11, r) = P (12, r − 1) find r.

Answer 1

P(11, r) = P (12, r − 1)

\[\Rightarrow \frac{11!}{\left( 11 - r \right)!} = \frac{12!}{\left( 13 - r \right)!}\]
\[ \Rightarrow \frac{\left( 13 - r \right)}{\left( 11 - r \right)!} = \frac{12!}{11!}\]
\[ \Rightarrow \frac{\left( 13 - r \right)\left( 12 - r \right)\left( 11 - r \right)!}{\left( 11 - r \right)!} = \frac{12 \times 11!}{11!}\]
\[ \Rightarrow \left( 13 - r \right)\left( 12 - r \right) = 12\]
\[ \Rightarrow \left( 13 - r \right)\left( 12 - r \right) = 4 \times 3\]
\[\text{On comparing the two sides, we get}: \]
\[13 - r = 4\]
\[ \Rightarrow r = 9\]