The number of ways of arranging p numbers out of 1, 2, 3, …, q so that maximum is q–2 and minimumis 2 (repetition of number is allowed) such that maximum and minimum both occur exactly once, (p > 5, q > 3) is

First we take one of the numbers as 2 and one another as q –2. We can arrange these two numbers inp(p–1) ways. We have to choose remaining p– 2 numbersfrom the numbers 3, 4, …, q – 4, q – 3. This can be done in(q−5)p−2 ways Thus, the total number of ways of arranging the numbersin de sired way isp(p−1)(q−5)p−2