Multinomial coefficient notation?
How does one evaluate this
$$\binom{5}{2, 2, 1}$$
would it be something like $\dfrac{5!}{2! \cdot 2! \cdot 1!}$ or does it evaluate differently then the usual $$\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$$ can someone explain?
$\endgroup$ 21 Answer
$\begingroup$You are right. An expression like $\displaystyle \binom{n}{k_1,k_2,k_3.....,k_l}$ is equal to $\displaystyle \frac{n!}{k_1!.k_2!.....k_l!}$
You can also write $\displaystyle \binom{n}{k}$ as $\displaystyle \binom{n}{k,n-k}$ but as this is used so often so the first one is preferred.
If you go to multinomial case then the coefficients will be somewhat like this i.e
$\displaystyle \left(\sum_{i=1}^{m}\alpha_i\right)^n=\sum_{\sum_{i=1}^m k_i=n}^{}\binom{n}{k_1,k_2,k_3.....,k_m}\prod_{i=1}^{m}\alpha_i^{k_i}$
Regarding understanding the notation: $\displaystyle \binom{n}{k_1,k_2,k_3.....,k_l}$ is Chosing $k_1$ objects from a collection of $n$ objects follwed by choosing $k_2$ objects from the rest $(n-k_1) $ objects and so on. Lastly $k_l $ objects from the remainint $(n-\sum_{i=1}^{l-1}k_i)$ objects.
$\endgroup$ 5
