Question

The number of ordered pairs of positive integers $(m,n)$ satisfying $m\le 2n\le 60,n\le 2m\le 60$ is

• A. 240
• B. 480
• C. 960
• D. none of these

Answer 1

Answer: (B)

Given $m\le 30,n\le 30$
$\Rightarrow$ Total cases $=30\times 30=900$
[Required condition: $2n\ge m,2m\ge n$
Let us find ordered pairs $(m,n)$ such that $2n<m,2m<n$.
By symmetry we will get same answer for both conditions. Hence, let us evaluate only one $2m<n$

Value of $n$	No. of points $(m,n)$
1,2	0
3,4	1
5,6	2
...
29,30	14

$\therefore$ Required number $=900-2\times \sum _{r=0}^{14}r=480$