Find r if ^5 P_r = 2^6 P_ r - 1 - Vidyadip

Question

Find r if $^5{P_r} = {2^6}{P_{r - 1}}$

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Answer

Here $^5{P_r} = {2^6}{P_{r - 1}}$
$\frac{{5!}}{{(5 - r)!}} = 2 \cdot \frac{{6!}}{{(7 - r)!}}$
$ \Rightarrow \frac{{5!}}{{(5 - r)!}} = \frac{{2 \times 6 \times 5!}}{{(7 - r)(6 - r)(5 - r)!}}$
$ \Rightarrow 1 = \frac{{12}}{{(7 - r)(6 - r)}}$$ \Rightarrow {r^2} - 13r + 42 = 12$
$ \Rightarrow {r^2} - 13r + 30 = 0$. $ \Rightarrow {r^2} - 10r - 3r + 30 = 0$
$ \Rightarrow r(r - 10) - 3(r - 10) = 0$
$ \Rightarrow (r - 10)(r - 3) = 0$
$ \Rightarrow $ r = 10 or r = 3
Now r = 10 is not possible because r > n
Thus r = 3

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