The binomial theorem gives the values of the coefficients of the expansion of:

where is any positive integer. (Newton gave the formula for any rational .)

The picture above shows the expansions for and .

The coefficients are the same as the rows of Pascal’s triangle.

is multiplied out by choosing one number from each bracket.

There is one way of choosing — choosing in each bracket, similarly one way of choosing .

There are ways of choosing — one for each bracket the is chosen from.

These are combinations of elements, (in the strict mathematical sense of “combination”,) in which identical elements and identical elements occur, where .

The number of them is given by .

( is defined as , so that the formula gives the right answer for the number of occurrences of and .)

So now we have the binomial expansion:

For example,

is usually written