Binomial expansion

The binomial expansion states that #m/thm/num

where the so-called binomial coëfficients are given by

and is the number of ways to choose elements of a set of size . See also Generalized binomial coëfficient.

Proof

#missing/proof

Properties

Proof of 1–3, 5

Clearly choosing elements from a set of size is the same as choosing elements to be excluded, proving ^P1.

Consider choosing a team of size from a set of people, where one member of the team is the captain. One can either first choose a captain and then the rest of the team (LHS), or the team and thence the captain (RHS), proving ^P2.

Consider a set of red marbles and blue marbles. The number of arbitrary choices of marbles is the LHS, but this is the same as every possible way of choosing red marbles and blue marbles (RHS). This proves ^P3.

A proof of ^P4 is missing, but ^P5 follows directly for .

Proof of 4

#missing/proof

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