Question

1) Qual o valor da combinação C 10,3? 2) Encontre o valor dos números binomiais:

Answer 1

Explicação passo-a-passo:

De modo geral:

$\sf \dbinom{n}{k}=\dfrac{n!}{k!\cdot(n-k)!}$

1)

$\sf C_{10,3}=\dbinom{10}{3}$

$\sf \dbinom{10}{3}=\dfrac{10!}{3!\cdot(10-3)!}$

$\sf \dbinom{10}{3}=\dfrac{10!}{3!\cdot7!}$

$\sf \dbinom{10}{3}=\dfrac{10\cdot9\cdot8\cdot7!}{6\cdot7!}$

$\sf \dbinom{10}{3}=\dfrac{720}{6}$

$\sf \dbinom{10}{3}=120$

$\sf C_{10,3}=120$

2)

a)

$\sf \dbinom{5}{2}=\dfrac{5!}{2!\cdot(5-2)!}$

$\sf \dbinom{5}{2}=\dfrac{5!}{2!\cdot3!}$

$\sf \dbinom{5}{2}=\dfrac{120}{2\cdot6}$

$\sf \dbinom{5}{2}=\dfrac{120}{12}$

$\sf \dbinom{5}{2}=10$

b)

$\sf \dbinom{10}{6}=\dfrac{10!}{6!\cdot(10-6)!}$

$\sf \dbinom{10}{6}=\dfrac{10!}{6!\cdot4!}$

$\sf \dbinom{10}{6}=\dfrac{10\cdot9\cdot8\cdot7\cdot6!}{6!\cdot24}$

$\sf \dbinom{10}{6}=\dfrac{5040}{24}$

$\sf \dbinom{10}{6}=210$

c)

$\sf \dbinom{8}{0}=\dfrac{8!}{0!\cdot(8-0)!}$

$\sf \dbinom{8}{0}=\dfrac{8!}{0!\cdot8!}$

$\sf \dbinom{8}{0}=1$