Let $c_1,\dots,c_n, s$ be $n+1$ real numbers. Compute:
$$I(c_1\dots,c_n;s) = \int_0^1 dx_1 \dots \int_0^1 dx_n \delta\left( \sum_{i=1}^n c_i x_i - s \right)$$
where $\delta(x)$ is Dirac delta function.
Disclaimer: This is not homework. I do not know if a closed form solution exists.