Question

Calcule o desenvolvimento dos números binomiais abaixo: ALGUÉM ME AJUDA POR FAVOR

Answer 1

Explicação passo-a-passo:

a)

$\sf \dbinom{7}{2}=\dfrac{7!}{2!\cdot(7-2)!}$

$\sf \dbinom{7}{2}=\dfrac{7!}{2!\cdot5!}$

$\sf \dbinom{7}{2}=\dfrac{7\cdot6\cdot5!}{2!\cdot5!}$

$\sf \dbinom{7}{2}=\dfrac{7\cdot6}{2!}$

$\sf \dbinom{7}{2}=\dfrac{42}{2}$

$\sf \dbinom{7}{2}=21$

b)

$\sf \dbinom{6}{3}=\dfrac{6!}{3!\cdot(6-3)!}$

$\sf \dbinom{6}{3}=\dfrac{6!}{3!\cdot3!}$

$\sf \dbinom{6}{3}=\dfrac{6\cdot5\cdot4\cdot3!}{3!\cdot3!}$

$\sf \dbinom{6}{3}=\dfrac{6\cdot5\cdot4}{3!}$

$\sf \dbinom{6}{3}=\dfrac{120}{6}$

$\sf \dbinom{6}{3}=20$

c)

Pela relação de Stifel:

$\sf \dbinom{n}{p-1}+\dbinom{n}{p}=\dbinom{n+1}{p}$

$\sf \dbinom{9}{5}+\dbinom{9}{6}=\dbinom{10}{6}$

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

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

$\sf =\dfrac{10\cdot9\cdot8\cdot7\cdot6!}{6!\cdot4!}$

$\sf =\dfrac{10\cdot9\cdot8\cdot7}{4!}$

$\sf =\dfrac{5040}{24}$

$\sf =210$

d)

$\sf \dbinom{12}{2m-1}=\dbinom{12}{m+4}$

• $\sf 2m-1=m+4$

$\sf 2m-m=4+1$

$\sf m=5$

• $\sf 2m-1+m+4=12$

$\sf 3m+3=12$

$\sf 3m=12-3$

$\sf 3m=9$

$\sf m=\dfrac{9}{3}$

$\sf m=3$