The numbers (read as choose ) are known as the binomial coefficients.
This is why is called the binomial coefficient.
[Pascal's triangle by Kazukiokumura]
Source: Figure 1.20, Bertsekas and Tsitsiklis (2008).
From the figure above, we can derive the following property of the binomial coefficient:
A combination is a choice of elements out of an -element set without regard to order.
A combination can be viewed as a partition of the set in two: one part contains element and the other part contains the remaining . We generalize by considering partitions into subsets, , whose sum is equal to .
The multinomial coefficient is:
which is equal to
collect terms, we get: