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Existence of Limit

Let Pn denote...

Question

Let $P_{n}$ denote the number of ways in which three people can be selected out of n people sitting in a row, if no two of them are consecutive. If, $P_{n + 1} - P_{n} = 15$ , then the value of n, is___.

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Solution

$P_{n} =^{n - 2} C_{3} a n d P_{n + 1} =^{n - 1} C_{3} ∴^{n - 1} C_{3} -^{n - 2} C_{3} = 15 \Rightarrow^{n - 2} C_{2} +^{n - 2} C_{3} -^{n - 2} C_{3} = 15 \Rightarrow^{n - 2} C_{2} = 15 \Rightarrow \frac{(n - 2)!}{2! (n - 4)!} = 15 \Rightarrow (n - 2) (n - 3) = 30 \Rightarrow n = 8$

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