First slide

Combinations

Question

Let P_ndenotes 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 ____________ .

Moderate

Solution

Here $P_{n} =^{n - 2} C_{3}$ and $P_{n + 1} =^{n - 1} C_{3}$
Hence, $^{n - 1} C_{3} -^{n - 2} C_{3} = 15$
or $^{n - 2} C_{3} +^{n - 2} C_{2} -^{n - 2} C_{3} = 15$
or $^{n - 2} C_{2} = 15$ cut $n = 8$

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