№329 Вычислить: 1) A^3_8 + P_3; 2) C^5_6 - A^6_6 + P_6; 3) A^2_{11} + A^4_{19}; 4) C^{11}_{16} - C^9_{11}

Объяснение:
Решим каждое выражение, используя формулы перестановок (P_n), размещений (A^k_n) и сочетаний (C^k_n).

Решение:

1. A^3_8 + P_3
   \[ A^3_8 = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336 \]
   \[ P_3 = 3! = 3 \times 2 \times 1 = 6 \]
   \[ A^3_8 + P_3 = 336 + 6 = 342 \]
2. C^5_6 - A^6_6 + P_6
   \[ C^5_6 = \frac{6!}{5!(6-5)!} = \frac{6!}{5!1!} = \frac{6}{1} = 6 \]
   \[ A^6_6 = P_6 = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
   \[ P_6 = 6! = 720 \]
   \[ C^5_6 - A^6_6 + P_6 = 6 - 720 + 720 = 6 \]
3. A^2_{11} + A^4_{19}
   \[ A^2_{11} = \frac{11!}{(11-2)!} = \frac{11!}{9!} = 11 \times 10 = 110 \]
   \[ A^4_{19} = \frac{19!}{(19-4)!} = \frac{19!}{15!} = 19 \times 18 \times 17 \times 16 = 93024 \]
   \[ A^2_{11} + A^4_{19} = 110 + 93024 = 93134 \]
4. C^{11}_{16} - C^9_{11}
   \[ C^{11}_{16} = \frac{16!}{11!(16-11)!} = \frac{16!}{11!5!} = \frac{16 \times 15 \times 14 \times 13 \times 12}{5 \times 4 \times 3 \times 2 \times 1} = \frac{524160}{120} = 4368 \]
   \[ C^9_{11} = \frac{11!}{9!(11-9)!} = \frac{11!}{9!2!} = \frac{11 \times 10}{2 \times 1} = 11 \times 5 = 55 \]
   \[ C^{11}_{16} - C^9_{11} = 4368 - 55 = 4313 \]

Ответ: 1) 342; 2) 6; 3) 93134; 4) 4313