First slide

Permutations

Question

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

Moderate

A

$^{p - 3} C_{q - 2}$
B

$^{p} C_{2} (q - 3)^{q - 1}$
C

$^{p} C_{2} \times^{q} C_{3}$
D

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

Solution

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

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