Let \(I_{n}=\int_{0}^{1}(1-x)^{n} \mathrm{e}^{x} \mathrm{~d} x\). Show that, for all positive integers \(n\),
\(I_{n}=n I_{n-1}-1\)

Find the exact value of \(I_{4}\).

By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y=(1-x)^{4} \mathrm{e}^{x}\) in the interval \(0 \leqslant x \leqslant 1\), show that
\(\frac{65}{24}\lt \mathrm{e}\lt \frac{11}{4} .\)