It is given that, for non-negative integers \(n\),
$$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$
Prove that, for \(n \geqslant 1\),
$$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
Evaluate \(I _ { 3 }\), giving the answer in terms of e.