We know that the product of r consecutive integers are divisible by r!.
The numerator consists of a1+a2+....+an consecutive integers.
The first set of a1 consecutive integers are divisible by a1!.
The second set of a2 consecutive integers are divisible by a2!.
...
The nth set of an consecutive integers are divisible by an!.
Hence the numerator is divisible by a1!*a2!*a3!*......*an!
Hence the given number is an integer.
It can also be said that the given number is the coefficient of x1a1*x2a2.....xnan of the expansion of :
(x1+x2+.....xn)a1+a2+a3...+an which is always an integer.