The number of ways of arranging p numbers out of$1, 2, 3, \dots , q$ so that maximum is$q–2$ and minimum
is 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 in
$p(p–1)$ ways. We have to choose remaining $p– 2$ numbers
from the numbers $3, 4, \dots , q – 4, q – 3.$ This can be done in

$(q-5{)}^{p-2}$ ways Thus, the total number of ways of arranging the numbers
in de sired way is

$p(p-1)(q-5{)}^{p-2}$