Multinomial Coefficient expressed as Product of Binomial Coefficients

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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 $\map P m$ 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

$\map P 2$ is the case:

\(\ds \dbinom {k_1 + k_2} {k_1, k_2}\) \(=\) \(\ds \frac {\paren {k_1 + k_2}!} {k_1! \, k_2!}\) Definition of Multinomial Coefficient
\(\ds \) \(=\) \(\ds \frac {\paren {k_1 + k_2}!} {k_1! \, \paren {\paren {k_1 + k_2} - k_1}!}\)
\(\ds \) \(=\) \(\ds \dbinom {k_1 + k_2} {k_1}\) Definition of Binomial Coefficient


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $\map P r$ is true, where $r \ge 2$, then it logically follows that $\map P {r + 1}$ 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:

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


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


Therefore:

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

$\blacksquare$


Sources

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