Given that a & b are both positive integers. For \(\frac{a!}{b!}\) to be integer a must be greater than or equal to b. otherwise a!/b! is going to be a fraction. We can then re-phrase the question to

Is a >= b?

Statement 1: (b-a)(b+a) = 7!+1. 7!+1 is +ve. b+a is +ve. b-a also has to be positive. note that b != a. then b>a. so a!/b! can not be integer.

Sufficient.

Statement 2: a+b=11. a=1 b=10 ++> a!/b! is not integer. reverse the values, that is b=1, a=10 then yes. Since we have a yes and no.

Insufficient.A is answer.

_________________

Click on Kudos if you liked the post!

Practice makes Perfect.