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Find r, if (i) ^5P_r = 2 ^6P_{r-1} (ii) ^5P_r= ^6 P_{r-1}

Question

Find $r$ , if
(i) $^{5} P_{r} = 2^{6} P_{r - 1}$
(ii) $^{5} P_{r} =^{6} P_{r - 1}$

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Owen Discussed

Find

r

, if
(i)

^{5} P_{r} = 2^{6} P_{r - 1}

(ii)

^{5} P_{r} =^{6} P_{r - 1}

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Text solutionVerified

(i)

^{5} P_{r} = 2^{6} P_{r - 1} \Rightarrow \frac{5 !}{( 5 - r )!} = 2 \cdot \frac{6 !}{[ 6 - ( r - 1 )]!}

\Rightarrow \frac{1}{( 5 - r )!} = 2 \cdot \frac{6}{( 7 - r )!} = \frac{12}{( 7 - r ) ( 6 - r ) [( 5 - r )]!}

\Rightarrow (7 - r) (6 - r) = 12 \Rightarrow r = 3

(ii)

^{5} P_{r} =^{6} P_{r - 1} \Rightarrow \frac{5 !}{( 5 - r )!} = \frac{6 !}{[ 6 - ( r - 1 )]!}

\Rightarrow \frac{1}{( 5 - r )!} = \frac{6}{( 7 - r )!} = \frac{6}{( 7 - r ) ( 6 - r ) [( 5 - r )]!}

\Rightarrow (7 - r) (6 - r) = 6 \Rightarrow r = 4

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