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The number of ways of arranging \mathrm{p} numbers out of \mathrm{1,2,3, \ldots, q} so that maximum is \mathrm{q-2} and minimum is 2 (repetition of number is allowed) such that maximum and minimum both occur exactly once, \mathrm{(p>5, q>3 )} is

Option: 1

\mathrm{{ }^{p-3} C_{q-2}}


Option: 2

\mathrm{{ }^{P} C_{2}(q-3)^{q-1}}


Option: 3

\mathrm{{ }^{p} C_{2} \times{ }^{q} C_{3}}


Option: 4

\mathrm{p(p-1)(q-5)^{p-2}}


Answers (1)

best_answer

First we take one of the numbers as 2 and one another as \mathrm{q-2}. We can arrange these two numbers in \mathrm{p(p-1)} ways. We have to choose remaining \mathrm{p-2}  numbers from the numbers \mathrm{3,4, \ldots, q-4, q-3}. This can be done in \mathrm{(q-5)^{p-2}} ways.
Thus, the total number of ways of arranging the numbers in desired way is \mathrm{p(p-1)(q-5)^{p-2}}.

Posted by

Ajit Kumar Dubey

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