Q. The number of ordered pairs of positive integers $(m, n)$ satisfying $m \leq 2 n \leq 60, n \leq 2 m \leq 60$ is
Permutations and Combinations
Solution:
Given $m \leq 30, n \leq 30$
$\Rightarrow $ Total cases $=30 \times 30=900$
[Required condition: $2 n \geq m, 2 m \geq n$
Let us find ordered pairs $(m, n)$ such that $2 n< m, 2 m < n$.
By symmetry we will get same answer for both conditions. Hence, let us evaluate only one $2 m < n$
Value of $n$
No. of points $(m, n)$
1,2
0
3,4
1
5,6
2
...
29,30
14
$\therefore $ Required number $=900-2 \times \displaystyle\sum_{r=0}^{14} r=480$
| Value of $n$ | No. of points $(m, n)$ |
|---|---|
| 1,2 | 0 |
| 3,4 | 1 |
| 5,6 | 2 |
| ... | |
| 29,30 | 14 |