P(B) = P( \bigcup_{i} A_i) = (Sylvester) = n \cdotp P(A_1) - \left(\begin {array}{c}n\\2 \end{array}\right) \cdotp P(A_1\bigcap A_2) + \left(\begin {array}{c}n\\3 \end{array}\right) \cdotp P (A_1\bigcap A_2\bigcap A_3) - ... + (-1)^{n-1} \left(\begin {array}{c}n\\n \end{array}\right) \cdotp P(\bigcap _i A_i) = n \cdotp \frac {1}{n} - \frac {n(n-1)}{2!} \cdotp \frac {1}{n(n-1)} + \frac {n(n-1)(n-2)}{3!} \cdotp \frac {1}{n(n-1)(n-2)} + ... + (-1)^{n-1} \cdotp \frac {1}{n!} = 1 - \frac {1}{2!} + \frac {1}{3!} - \frac {1}{4!} + ... + (-1)^{n-1} \cdotp \frac {1}{n!} = \sum _{k=1}^n (-1)^{k-1} \cdotp \frac {1}{k!}