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:
is usually written