Упр.1090 ГДЗ Алимов 10-11 класс (Алгебра)

1) C 10/13 + C 11/13;
2) C 12/14 + C 13/14;
3) C 4/19 — C 4/18;
4) C 3/21 — C 3/20;
5) C 3/61 — C 2/60;
6) C 3/71 + C 2/70.

Используем рекуррентные свойства сочетаний:

$$C_m^n + C_m^{n+1} = C_{m+1}^{n+1},$$

$$C_m^{n+1} — C_m^n = C_{m-1}^n.$$

1. $$C_{13}^{10} + C_{13}^{11} = C_{14}^{11}$$

   $$C_{14}^{11} = \frac{14!}{11!\,3!} = \frac{14\cdot 13\cdot 12}{3\cdot 2} = 364$$

2. $$C_{14}^{12} + C_{14}^{13} = C_{15}^{13}$$

   $$C_{15}^{13} = \frac{15!}{13!\,2!} = \frac{15\cdot 14}{2} = 105$$

3. $$C_{19}^{4} — C_{18}^{4} = C_{18}^{3}$$

   $$C_{18}^{3} = \frac{18!}{15!\,3!} = \frac{18\cdot 17\cdot 16}{3\cdot 2} = 816$$

4. $$C_{21}^{3} — C_{20}^{3} = C_{20}^{2}$$

   $$C_{20}^{2} = \frac{20!}{18!\,2!} = \frac{20\cdot 19}{2} = 190$$

5. $$C_{61}^{3} — C_{60}^{2} = C_{60}^{3}$$

   $$C_{60}^{3} = \frac{60!}{57!\,3!} = \frac{60\cdot 59\cdot 58}{3\cdot 2} = 34220$$

6. $$C_{71}^{3} — C_{70}^{2} = C_{70}^{3}$$

   $$C_{70}^{3} = \frac{70!}{67!\,3!} = \frac{70\cdot 69\cdot 68}{3\cdot 2} = 54740$$

Ответ

1) $$364$$; 2) $$105$$; 3) $$816$$; 4) $$190$$; 5) $$34220$$; 6) $$54740$$.