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:
- 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 $=5 \times 9 \times 5=225$
