Product of r Choose m with m Choose k
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Theorem
Let $r \in \R, m \in \Z, k \in \Z$.
Then:
- $\dbinom r m \dbinom m k = \dbinom r k \dbinom {r - k} {m - k}$
where $\dbinom r m$ is a binomial coefficient.
Complex Numbers
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers:
- $\dbinom z t \dbinom t w = \dbinom z w \dbinom {z - w} {t - w}$
where $\dbinom z w$ is a binomial coefficient.
Proof 1
Integral Index
Let $r \in \Z$.
Then:
\(\ds \binom r m \binom m k\) | \(=\) | \(\ds \frac {r^{\underline m} } {m!} \frac {m^{\underline k} } {k!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {r! m!} {m! \paren {r - m}! k! \paren {m - k}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {r! \paren {r - k}!} {k! \paren {r - k}! \paren {m - k}! \paren {r - m}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \binom r k \binom {r - k} {m - k}\) |
$\Box$
Real Index
Both sides of the above equation are polynomials in $r$.
Since these polynomials agree for all $r \in \Z$, they must agree for all $r \in \R$.
$\blacksquare$
Proof 2
Consider the trinomial coefficient:
- $\dbinom r {k, m - k, r - m}$
We use Multinomial Coefficient expressed as Product of Binomial Coefficients:
- $\dbinom {k_1 + k_2 + k_3} {k_1, k_2, k_3} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2}$
and substitute as appropriate for $k_1, k_2, k_3$.
We have:
\(\ds \) | \(\) | \(\ds \dbinom r {k, m - k, r - m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {\left({m - k}\right) + \left({r - m}\right) + k} {m - k, r - m, k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {\left({m - k}\right) + \left({r - m}\right)} {m - k} \dbinom {\left({m - k}\right) + \left({r - m}\right) + k} {\left({m - k}\right) + \left({r - m}\right)}\) | Multinomial Coefficient expressed as Product of Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {r - k} {m - k} \dbinom r m\) |
Similarly:
\(\ds \) | \(\) | \(\ds \dbinom r {k, m - k, r - m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {k + \left({m - k}\right) + \left({r - m}\right)} {k, m - k, r - m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom {k + \left({m - k}\right)} k \dbinom {k + \left({m - k}\right) + \left({r - m}\right)} {\left({m - k}\right) + k}\) | Multinomial Coefficient expressed as Product of Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \dbinom m k \dbinom r m\) |
$\blacksquare$
The result follows on equating the two expressions.