Question

Com calculos:
Calcule o valor da soma binomial
(12/3)+(12/4)+(13/5)+(14/6)

Answer 1

Oie, Td Bom?!

= (12 3) + (12 4) + (13 5) + (14 6)

• Desenvolvendo as expressões usando (n k) = n!/[k! . (n - k)!].

= 12!/[3! . (12 - 3)!] + 12![4! . (12 - 4)!] + 13!/[5! . (13 - 5)!] + 14!/[6! . (14 - 6)!]

= Desnolva as expressões usando n! = n . (n - 1)! e subtraia os números.

= [12 . 11 . 10 . 9!]/[3! . 9!] + [12 . 11 . 10 . 9 . 8!]/[4! . 8!] + [13 . 12 . 11 . 10 . 9 . 8!]/[5! . 8!] + [14 . 13 . 12 . 11 . 10 . 9 . 8!]/[6! . 8!]

= [12 . 11 . 10]/3! + [12 . 11 . 10 . 9]/4!+ [13 . 12 . 11 . 10 . 9]/5! + [14 . 13 . 12 . 11 . 10 . 9]/6!

= 1320/6 + 11880/24 + 154440/120 + 2162160/720

= 220 + 495 + 1287 + 3003

= 220 + 495 + 4290

= 4510 + 495

= 5005

Att. Makaveli1996

Answer 2

Explicação passo-a-passo:

Seja $\sf S=\dbinom{12}{3}+\dbinom{12}{4}+\dbinom{13}{5}+\dbinom{14}{6}$

Pela relação de Stifel:

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

Assim:

$\sf \dbinom{12}{3}+\dbinom{12}{4}=\dbinom{13}{4}$

Então:

$\sf S=\dbinom{13}{4}+\dbinom{13}{5}+\dbinom{14}{6}$

Novamente pela relação de Stifel:

$\sf \dbinom{13}{4}+\dbinom{13}{5}=\dbinom{14}{5}$

Desse modo:

$\sf S=\dbinom{14}{5}+\dbinom{14}{6}$

Pela relação de Stifel:

$\sf \dbinom{14}{5}+\dbinom{14}{6}=\dbinom{15}{6}$

Logo:

$\sf S=\dbinom{15}{6}$

$\sf S=\dfrac{15\cdot14\cdot13\cdot12\cdot11\cdot10}{6!}$

$\sf S=\dfrac{3603600}{720}$

$\sf S=5005$