Find: r, if 5 ^4P_r = 6 ^5P_ r - 1 - Vidyadip

Question

Find: $r$, if $5 ^4P_r = 6 ^5P_{r - 1}$

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Answer

We have, $5. ^4P_r = 6 . (^5P_{r - 1})$
$ \Rightarrow 5 \cdot \frac{4 !}{(4-r) !}=6 \times \frac{5 !}{[5-(r-1)] !} $
$ \Rightarrow \frac{5 \cdot 4 !}{(4-r) !}=\frac{6 \times 5 \times 4 !}{(6-r) !} $
$ \Rightarrow \quad \frac{1}{(4-r) !}=\frac{6}{(6-r)(5-r)(4-r) !} $
$\Rightarrow (6 - r) (5 - r) = 6$
$\Rightarrow 30 - 11r + r^2 = 6$
$\Rightarrow r^2 - 11r + 24 = 0$
$\Rightarrow (r - 3) (r - 8) = 0$
$\Rightarrow r = 3, 8$
But r $ \neq $ 8, because in $^4P_r, r$ cannot be greater than $4$.
Hence, $r = 3$

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