Question

me ajudem por favor

Answer 1

De modo geral,

$A_{r,s}=\dfrac{r!}{(r-s)!}$

Assim:

a) $A_{12,3}=\dfrac{12!}{(12-3)!}=\dfrac{12!}{9!}$

Veja que, $12!=12\cdot11\cdot10\cdot9!$, então:

$A_{12,3}=\dfrac{12!}{9!}=\dfrac{12\cdot11\cdot10\cdot9!}{9!}=12\cdot11\cdot10$

$A_{12,3}=1~320$


b) $A_{6,3}=\dfrac{6!}{(6-3)!}=\dfrac{6!}{3!}=\dfrac{6\cdot5\cdot4\cdot3!}{3!}$

$A_{6,3}=6\cdot5\cdot4=120$


c) $A_{6,6}=\dfrac{6!}{(6-6)!}=\dfrac{6!}{0!}=6!=6\cdot5\cdot4\dots\cdot1=720$


d) $A_{15,3}=\dfrac{15!}{(15-3)!}=\dfrac{15!}{12!}=15\cdot14\cdot13=2~730$


De modo geral,

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

Logo:

e) $C_{10,3}=\dbinom{10}{3}=\dfrac{10!}{3!\cdot7!}$

Note que, $10!=10\cdot9\cdot8\cdot7!$, logo:

$C_{10,3}=\dfrac{10\cdot9\cdot8\cdot7!}{3!\cdot7!}=120$


f) $C_{16,7}=\dfrac{16!}{7!\cdot9!}=11~440$

g) $C_{8,8}=\dfrac{8!}{8!\cdot0!}=1$


h) $C_{8,0}=\dfrac{8!}{0!\cdot8!}=1$

Curiosidade:

Se $k+m=n$, então $\dbinom{n}{k}=\dbinom{n}{m}$

Exemplo: $\dbinom{8}{8}=\dbinom{8}{0}$, pois $0+8=8$.