P(A_1\cup\cdots\cup A_n)=\sum_{k=1}(-1)^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}P(A_{i_1}\cdots A_{i_k})=\sum_{k=1}^n(-1)^{k-1}{n\choose k}\frac{(n-k)!}{n!}=n!\sum_{k=1}^n(-1)^{k-1}