Упр.455 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) C15 12 + C15 13;
2) C11 3 + C11 2;
3) C21 4 — C20 4;
4) C101 3 — C100 3.

1. $$C_{15}^{12}+C_{15}^{13}=C_{15}^{12}+C_{15}^{15-12}=C_{16}^{13}$$
   по формуле Паскаля.

   $$C_{16}^{13}=C_{16}^{3}=\frac{16!}{3!\,13!}=\frac{16\cdot 15\cdot 14}{3\cdot 2}=560.$$

2. $$C_{11}^{3}+C_{11}^{2}=C_{11}^{2+1}+C_{11}^{2}=C_{12}^{3}.$$

   $$C_{12}^{3}=\frac{12!}{3!\,9!}=\frac{12\cdot 11\cdot 10}{3\cdot 2}=220.$$

3. $$C_{21}^{4}-C_{20}^{4}=C_{20}^{3+1}-C_{20}^{4}=C_{20}^{3}.$$

   $$C_{20}^{3}=\frac{20!}{3!\,17!}=\frac{20\cdot 19\cdot 18}{3\cdot 2}=1140.$$

4. $$C_{101}^{3}-C_{100}^{3}=C_{100}^{2+1}-C_{100}^{3}=C_{100}^{2}.$$

   $$C_{100}^{2}=\frac{100!}{2!\,98!}=\frac{100\cdot 99}{2}=4950.$$

Ответ

1) $$560$$; 2) $$220$$; 3) $$1140$$; 4) $$4950$$.