Упр.488 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

С помощью свойств числа сочетаний найти:

1. 1) $$C_{12}^{2}+C_{12}^{3}+C_{13}^{4}$$;
   2) $$C_{9}^{3}+C_{9}^{4}+C_{10}^{5}$$;
   3) $$C_{8}^{4}-C_{7}^{4}$$;
   4) $$C_{9}^{3}-C_{8}^{2}$$;
   5) $$C_{7}^{4}+C_{7}^{5}+C_{7}^{6}+C_{7}^{7}$$;
   6) $$C_{5}^{0}+C_{5}^{1}+C_{5}^{2}+C_{9}^{9}+C_{9}^{8}+C_{9}^{7}+C_{9}^{6}+C_{9}^{5}$$.

1. $$C_{12}^{2}+C_{12}^{3}+C_{13}^{4}=C_{13}^{3}+C_{13}^{4}=C_{14}^{4}=1001.$$
2. $$C_{9}^{3}+C_{9}^{4}+C_{10}^{5}=C_{10}^{4}+C_{10}^{5}=C_{11}^{5}=462.$$
3. $$C_{8}^{4}-C_{7}^{4}=C_{7}^{3}+C_{7}^{4}-C_{7}^{4}=C_{7}^{3}=35.$$
4. $$C_{9}^{3}-C_{8}^{2}=C_{8}^{3}+C_{8}^{2}-C_{8}^{2}=C_{8}^{3}=56.$$
5. $$C_{7}^{4}+C_{7}^{5}+C_{7}^{6}+C_{7}^{7}=C_{7}^{3}+C_{7}^{4}+C_{7}^{5}+C_{7}^{6}+C_{7}^{7}$$
   $$=C_{8}^{4}=64.$$
6. $$C_{5}^{0}+C_{5}^{1}+C_{5}^{2}+C_{9}^{9}+C_{9}^{8}+C_{9}^{7}+C_{9}^{6}+C_{9}^{5}$$
   $$=(C_{5}^{0}+C_{5}^{1}+C_{5}^{2})+(C_{9}^{9}+C_{9}^{8}+C_{9}^{7}+C_{9}^{6}+C_{9}^{5})$$
   $$=2^{5-1}+2^{9-1}=16+256=272.$$

Ответ

1) 1001; 2) 462; 3) 35; 4) 56; 5) 64; 6) 272.