= Binomial Theorem =

The '''binomial theorem''' is a method for expanding binomial expression.

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== Description ==

Consider the binomial expression ''(a + b)^n^''. It expands to a sum of terms that are all of the form ''a^n-k^b^k^''. (Or equivalently, ''a^k^b^n-k^''.) There are always ''n+1'' different forms of terms in the expression.

The number of terms that are of the form ''a^n-k^b^k^'' for a specific value of ''k'' is given by a [[Analysis/Combinatorics#Combinations|combination]]. For example, consider ''(a + b)^2^''; the number of terms that are of the form ''a^0^b^2^ = b^2^'' is 1.

{{attachment:comb1.svg}}

The number of terms that are of the form ''a^2^b^0^ = a^2^'' is also 1.

{{attachment:comb2.svg}}

(It should be apparent that ''a^n-k^b^k^'' and ''a^k^b^n-k^'' are still equivalent.)

More generally, for a specific value of ''k'', the number of terms in the expansion of ''(a + b)^n^'' that are of the form ''a^n-k^b^k^'' is given by:

{{attachment:comb3.svg}}

(And the equivalence of ''a^n-k^b^k^'' and ''a^k^b^n-k^'' is now proven generally by Pascal's rule.)

The original binomial expression ''(a + b)^n^'' can therefore be solved as a sum over all values of ''k'' from zero to ''n''.

{{attachment:binom.svg}}

The number of terms of a specific form is a '''binomial coefficient'''.



=== Pascal's Triangle ===

Pascal's triangle is formed by rows of binomial coefficients; the ''n''th row (starting at 0) is composed of the binomial coefficients for the binomial expression.

Consider:
 * ''(a+b)^0^ = (1)''
 * ''(a+b)^1^ = (1)a + (1)b''
 * ''(a+b)^2^ = (1)a^2^ + (2)ab + (1)b^2^''
 * ''(a+b)^3^ = (1)a^3^ + (3)a^2^b +  (3)ab^2^ + (1)b^3^''

And compare this to:

{{attachment:pascal.svg}}



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