Permutations and Combinations 2 Question 12

12. If $r, s, t$ are prime numbers and $p, q$ are the positive integers such that LCM of $p, q$ is $r^{2} s^{4} t^{2}$, then the number of ordered pairs $(p, q)$ is

(2006, 3M)

(a) 252

(b) 254

(c) 225

(d) 224

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Answer:

Correct Answer: 12. (c)

Solution:

  1. Since, $r, s, t$ are prime numbers.

$\therefore$ Selection of $p$ and $q$ are as under

$\boldsymbol{p}$ $\boldsymbol{q}$ Number of ways
$r^{0}$ $r^{2}$ 1 way
$r^{1}$ $r^{2}$ 1 way
$r^{2}$ $r^{0}, r^{1}, r^{2}$ 3 ways

$\therefore$ Total number of ways to select, $r=5$

Selection of $s$ as under

$\therefore$ Total number of ways to select $s=9$ Similarly, the number of ways to select $t=5$

$\therefore$ Total number of ways $=5 \times 9 \times 5=225$



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