let m=sqrt(a-b + c-d) , if a,b,c,d are replaced by 1,2,3,4 with no repetition how many of m will be a real no?

Answer 1

there's 4! = 24 total combinations.

of these, we only like the ones where
b + d = 5, or b + d = 6, or b + d = 7
b+d=5: can use [1,4] and [2,3] pair, so 8 combinations
b+d=6: 4 combinations
b+d=7: 4 combinatios

so, for 16 combinations, m is a real number, for 8 remaining it is not.

Answer 2

m = √(a - b + c - d)
= √((a + c) - (b + d))

For m to be real, √((a + c) - (b + d)) ≥ 0 must be true

So you need (a + c) ≥ (b + d)

Since the numbers add to 10, a + c ≥ 5

So

a..c
4..3
4..2
4..1
3..4
3..2
2..4
2..3
1..4

8 options. However, in each case, there are two variations for b and d, so the answer is 8 * 2 = 16

Ans: 16