Hopefully someone will give you a more elegant soln than mine!

Assume that x is rational so the integers N and M exist such that:

x=N/M

Therefore, x*(1!2!3!.....M!) must be an integer.

But

Now the first M terms of this are integers. Define S to be the integer that is the sum of the first M integers then:

Now provided M > 1

Since this term is between 0 and 1 it is not an integer and we have a contradiction with our statement that N is an integer and hence x must be irrational.