It would be divisible by 5 only if (2 ^ m + 2 ^ n) will have a 0 or 5 in the last digit . Now 5 cannot occur as the last digit in this case since 2 even no addition can never result odd number .
Now the possible powers of 2 within 10 are = {2,4,8,16,32,64,128,256,512,1024}
So the possible combinations which will bring a 0 as the last digit is
{ ( 2,8) ,(4,16 ), (8,32) ,(16,64) , (128,2) , (512 ,8) , (1024,16 ) , (1024 , 256 ) , (512,128) , (256 ,4 ) ,(128 , 32) ,(256,64) }
There are 12 pairs . But since these are ordered pairs (a,b ) will be treated differently than (b,a ) .
Hence there are 12 * 2 = 24 such pairs.
Ans is option C.