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Question
If P (5, r) = P (6, r − 1), find r ?
Solution
P (5, r) = P (6, r − 1)
or 5Pr = 6Pr-1
\[\frac{5!}{\left( 5 - r \right)!} = \frac{6!}{\left( 6 - r + 1 \right)!}\]
\[ \Rightarrow \frac{\left( 6 - r + 1 \right)!}{\left( 5 - r \right)!} = \frac{6!}{5!}\]
\[ \Rightarrow \frac{(7 - r)!}{\left( 5 - r \right)!} = \frac{6\left( 5! \right)}{5!}\]
\[ \Rightarrow \frac{\left( 7 - r \right)\left( 6 - r \right)\left( 5 - r \right)!}{\left( 5 - r \right)!} = 6\]
\[ \Rightarrow \left( 7 - r \right)\left( 6 - r \right) = 6\]
\[ \Rightarrow \left( 7 - r \right)\left( 6 - r \right) = 3 \times 2\]
\[\text{On comparing the above two equations, we get}: \]
\[7 - r = 3\]
\[ \Rightarrow r = 4\]
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