Q. 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
IIT JEEIIT JEE 2006Permutations and Combinations
Solution:
Since, $r, s, t$ are prime numbers.
$\therefore$ Selection of $p$ and $q$ are as under
p
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
$s^0$
$s^4$
1 way
$s^1$
$s^4$
1 way
$s^2$
$s^4$
1 ways
$s^3$
$s^4$
1 ways
$s^4$
15 ways
$\therefore$ Total number of ways to select $s=9$
Similarly, the number of ways to select $t=5$
$\therefore$ Total number of ways $=5 \times 9 \times 5=225$
| p | 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 |
| $s^0$ | $s^4$ | 1 way |
| $s^1$ | $s^4$ | 1 way |
| $s^2$ | $s^4$ | 1 ways |
| $s^3$ | $s^4$ | 1 ways |
| $s^4$ | 15 ways |