Question

find n if p(10.n+1) : p(11 .n )=30:11

Answer 1

Hi...☺

Here is your answer...✌

$^{10} P_{n + 1} \ratio \: ^{11} P_{n } = \frac{30}{11} \\ \\ \frac{ ^{10} P_{n + 1}}{ ^{11} P_{n }} = \frac{30}{11 } \\ \\11 \times {}^{10} P_{n + 1} =30 \times {}^{11} P_{n} \\ \\ \frac{11 \times 10!}{(10 - (n + 1))!} = \frac{30 \times 11!}{(11- n )!} \\ \\ \frac{11!}{(9 - n) !} = \frac{30 \times11!} {(11 - n)!} \\ \\ \frac{1}{(9 - n) !} = \frac{30}{(11 - n) \times (10 - n) \times (9 - n) !} \\ \\ (11 - n)(10 - n) = 30 \\ \\ 110 - 11n - 10n + {n}^{2} - 30 = 0 \\ \\ \implies {n}^{2} - 21n + 80 = 0 \\ \\\implies {n}^{2} - 16n - 5n + 80 = 0 \\ \\ \implies n(n - 16) - 5(n - 16) = 0 \\ \\ \implies (n - 16)(n - 5) = 0 \\ \\ \implies n = 16 \: \: or \: \: n = 5 \\ \\ \implies n = 5 \\ \\ (by \: rejecting \: n = 16 \: as \: n \: cannot \: be \: \\ greater \: than \: 10)$

Answer 2

your answer is 5

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