Editing Binomial coefficient (section)

=== Products of binomial coefficients ===
One can express the product of two binomial coefficients as a linear combination of binomial coefficients:
: <math>{z \choose m} {z\choose n} = \sum_{k=0}^{\min(m,n)} {m + n - k \choose k, m - k, n - k} {z \choose m + n - k},</math>

where the connection coefficients are [[Multinomial theorem|multinomial coefficients]]. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign {{math|''m'' + ''n'' − ''k''}} labels to a pair of labelled combinatorial objects—of weight ''m'' and ''n'' respectively—that have had their first ''k'' labels identified, or glued together to get a new labelled combinatorial object of weight {{math|''m'' + ''n'' − ''k''}}. (That is, to separate the labels into three portions to apply to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what [[falling factorial]]s are to ordinary generating series.

The product of all binomial coefficients in the ''n''th row of the Pascal triangle is given by the formula:
: <math>\prod_{k=0}^{n}\binom{n}{k}=\prod_{k=1}^{n}k^{2k-n-1}.</math>