Multinomial Coefficient expressed as Product of Binomial Coefficients

Theorem

$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_m} {k_1 + k_2 + \cdots + k_{m - 1} }$

where:

$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m}$ denotes a multinomial coefficient
$\dbinom {k_1 + k_2} {k_1}$ etc. denotes binomial coefficients.

Proof

The proof proceeds by induction.

For all $m \in \Z_{> 1}$, let $P \left({m}\right)$ be the proposition:

$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_m} {k_1 + k_2 + \cdots + k_{m - 1} }$

Basis for the Induction

$P \left({2}\right)$ is the case:

 $\displaystyle \dbinom {k_1 + k_2} {k_1, k_2}$ $=$ $\displaystyle \frac {\left({k_1 + k_2}\right)!} {k_1! \, k_2!}$ $\quad$ Definition of Multinomial Coefficient $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\left({k_1 + k_2}\right)!} {k_1! \, \left({\left({k_1 + k_2}\right) - k_1}\right)!}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \dbinom {k_1 + k_2} {k_1}$ $\quad$ Definition of Binomial Coefficient $\quad$

This is the basis for the induction.

Induction Hypothesis

Now it needs to be shown that, if $P \left({r}\right)$ is true, where $r \ge 2$, then it logically follows that $P \left({r + 1}\right)$ is true.

So this is the induction hypothesis:

$\dbinom {k_1 + k_2 + \cdots + k_r} {k_1, k_2, \ldots, k_r} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_r} {k_1 + k_2 + \cdots + k_{r - 1} }$

from which it is to be shown that:

$\dbinom {k_1 + k_2 + \cdots + k_r + k_{r + 1} } {k_1, k_2, \ldots, k_r, k_{r + 1} } = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_r + k_{r + 1} } {k_1 + k_2 + \cdots + k_r}$

Induction Step

This is the induction step:

 $\displaystyle$  $\displaystyle \dbinom {k_1 + k_2 + \cdots + k_r + k_{r + 1} } {k_1, k_2, \ldots, k_r, k_{r + 1} }$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\left({k_1 + k_2 + \cdots + k_r + k_{r + 1} }\right)!} {k_1! \, k_2! \, \cdots k_r! \, k_{r + 1}!}$ $\quad$ Definition of Multinomial Coefficient $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\left({k_1 + k_2 + \cdots + k_r}\right)!} {k_1! \, k_2! \, \cdots k_r!} \frac {\left({k_1 + k_2 + \cdots + k_r + k_{r + 1} }\right)!} {\left({k_1 + k_2 + \cdots + k_r}\right)! k_{r + 1} }$ $\quad$ multiplying top and bottom by $\left({k_1 + k_2 + \cdots + k_r + k_{r + 1} }\right)!$ $\quad$ $\displaystyle$ $=$ $\displaystyle \dbinom {k_1 + k_2 + \cdots + k_r} {k_1, k_2, \ldots, k_r} \frac {\left({k_1 + k_2 + \cdots + k_r + k_{r + 1} }\right)!} {\left({k_1 + k_2 + \cdots + k_r}\right)! k_{r + 1} }$ $\quad$ Definition of Multinomial Coefficient $\quad$ $\displaystyle$ $=$ $\displaystyle \dbinom {k_1 + k_2 + \cdots + k_r} {k_1, k_2, \ldots, k_r} \dbinom {k_1 + k_2 + \cdots + k_r + k_{r + 1} } {k_1 + k_2 + \cdots + k_r}$ $\quad$ Definition of Binomial Coefficient $\quad$ $\displaystyle$ $=$ $\displaystyle \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_r} {k_1 + k_2 + \cdots + k_{r - 1} } \dbinom {k_1 + k_2 + \cdots + k_r + k_{r + 1} } {k_1 + k_2 + \cdots + k_r}$ $\quad$ Induction Hypothesis $\quad$ $\displaystyle$ $=$ $\displaystyle \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_r + k_{r + 1} } {k_1 + k_2 + \cdots + k_r}$ $\quad$ $\quad$

So $P \left({r}\right) \implies P \left({r + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\forall m \in \Z_{>1}: \dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_m} {k_1 + k_2 + \cdots + k_{m - 1} }$

$\blacksquare$