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  • The number of ordered pairs of positive integers   satisfying <img alt="m \leq 2 n

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

 

Option: 1

240


Option: 2

480


Option: 3

960

 


Option: 4

none of these


Answers (1)

best_answer

\begin{aligned} & \text { Given } m \leq 30, n \leq 30 \\ & \Rightarrow \text { Total cases }=30 \times 30=900 \\ & \text { [Required condition: } 2 n \geq m, 2 m \geq n \end{aligned}
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  2m<n

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

\therefore \text { Required number }=900-2 \times \sum_{r=0}^{14} r=480
 

Posted by

himanshu.meshram

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