If P (5, r) = P (6, r − 1), find r.

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Solution

P (5, r) = P (6, r − 1)
or ⁵P_r = ⁶P_r-1
$\frac{5!}{(5 - r)!} = \frac{6!}{(6 - r + 1)!} \Rightarrow \frac{(6 - r + 1)!}{(5 - r)!} = \frac{6!}{5!} \Rightarrow \frac{(7 - r)!}{(5 - r)!} = \frac{6 (5!)}{5!} \Rightarrow \frac{(7 - r) (6 - r) (5 - r)!}{(5 - r)!} = 6 \Rightarrow (7 - r) (6 - r) = 6 \Rightarrow (7 - r) (6 - r) = 3 \times 2 On comparing the above two equations, we get : 7 - r = 3 \Rightarrow r = 4$

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