Question

Calcule os números binomiais:

$\left \ ({{100} \atop {2}} )\right.$
$\left \ ({{100} \atop {98}} )\right.$

Answer 1

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

$\dbinom{100}{2}=\dfrac{100!}{2!\cdot(100-2)!}=\dfrac{100!}{2!\cdot98!}=\dfrac{100\cdot99\cdot98!}{2!\cdot98!}=\dfrac{100\cdot99}{2}=4~950$

Veja que, $\dbinom{n}{p}=\dbinom{n}{n-p}$. Como $\dbinom{100}{2}=4~950$ e $100-2=98$, temos:

$\dbinom{100}{98}=\dbinom{100}{2}=4~950$

Answer 2

$\left(_2^{100}\right)=\frac{100!}{(100-2)!2!}=\frac{100!}{98!2}=\frac{100*99*98!}{98!2}=\frac{100*99}{2}=4950$

$\left(_{98}^{100}\right)=\frac{100!}{(100-98)!2!}=4.950$