Find: r, if 5 4Pr = 6 5Pr - 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 + r² = 6
$\Rightarrow$ r² - 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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