1. Вычислите: а) \(\frac{5!}{2!}\); б) \(\frac{10!}{8!}\); в) \(\frac{15!}{13!-2!}\); г) \(C_8^2\); д) \(C_3^2 \cdot C_6^4\); e) \(C_2^2 - C_5^5\)

a) \(\frac{5!}{2!}\) = \(\frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}\) = \(5 \times 4 \times 3\) = 60

Ответ: 60

б) \(\frac{10!}{8!}\) = \(\frac{10 \times 9 \times 8!}{8!}\) = \(10 \times 9\) = 90

Ответ: 90

в) \(\frac{15!}{13! \cdot 2!}\) = \(\frac{15 \times 14 \times 13!}{13! \times 2 \times 1}\) = \(\frac{15 \times 14}{2}\) = \(15 \times 7\) = 105

Ответ: 105

г) \(C_8^2\) = \(\frac{8!}{2! \cdot (8-2)!}\) = \(\frac{8!}{2! \cdot 6!}\) = \(\frac{8 \times 7 \times 6!}{2! \cdot 6!}\) = \(\frac{8 \times 7}{2}\) = \(4 \times 7\) = 28

Ответ: 28

д) \(C_3^2 \cdot C_6^4\) = \(\frac{3!}{2! \cdot (3-2)!} \cdot \frac{6!}{4! \cdot (6-4)!}\) = \(\frac{3!}{2! \cdot 1!} \cdot \frac{6!}{4! \cdot 2!}\) = \(\frac{3 \times 2!}{2! \cdot 1} \cdot \frac{6 \times 5 \times 4!}{4! \cdot 2 \times 1}\) = \(3 \cdot \frac{6 \times 5}{2}\) = \(3 \cdot \frac{30}{2}\) = \(3 \cdot 15\) = 45

Ответ: 45

e) \(C_2^2 - C_5^5\) = \(\frac{2!}{2! \cdot (2-2)!} - \frac{5!}{5! \cdot (5-5)!}\) = \(\frac{2!}{2! \cdot 0!} - \(\frac{5!}{5! \cdot 0!}\) = \(\frac{2!}{2! \cdot 1} - \frac{5!}{5! \cdot 1}\) = \(1 - 1\) = 0

Ответ: 0