Urgently!!! Need to simplify multiple integral

Please, help:

Is it possible to simplify it?
I tried to simplify the numerator of fraction.
My result:
- is it true?

Also my result of denominator for equal functions, when is:

but I'm not sure ...
I want to solve this problem for case, when all these functions are different. Help ...
I'm considering the case when .

Thanks in advance.

If \(\displaystyle F(x)\) happens to be the density function of a continuous random variable, then your last integral has the value \(\displaystyle 1/n!\), which can be seen by a simple substitution.

F(x) is not a density function, F(x) is probability distribution function (0<F(x)<1).

Same result. In your last expression, let \(\displaystyle y = \int_x ^\infty F(t) dt\).