1.

If P (5,r) = 2 X P(6, r-1) find r

Answer»

Hi...☺

Here is your ANSWER....✌

GIVEN THAT,

{} ^{<klux>5</klux>}P_{r} = 2 \times {}^{6}P_{r - 1} \\ \\ = > \frac{5!}{(5 - r)!} = 2 \times \frac{6!}{(6 - (r - 1))!} \\ \\ \frac{5!}{(5 - r)!} = 2 \times \frac{6\times 5!}{(6 - r + 1)!} \\ \\ \frac{1}{(5 - r)!} = 2 \times \frac{6}{(7 - r)!} \\ \\ \frac{1}{(5 - r)!} = \frac{12}{(7 - r) \times (6 - r) \times (5 - r)!} \\ \\ 1 = \frac{12}{(7 - r) \times (6 - r) } \\ \\ (7 - r)(6 - r) = 12 \\ \\ 42 - 13r + r {}^{2} = 12 \\ \\ {r}^{2} - 13r + 30 = 0 \\ \\ {r}^{2} - 10r - 3r + 30 = 0 \\ \\ {r}(r - 10) - 3(r - 10) = 0 \\ \\ (r - 3)(r - 10) = 0 \\ \\ = > r = 3 \: \: or \: \: r = 10 \\ \\ => r = 3

SINCE,
Value of r can't be GREATER than 5

[ As we know r ≤ n ]

HENCE,
r = 3


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