## The Binomial Expansion

We can use the binomial theorem to expand and simplify expressionsof the formwhereisany positive whole number.

The general expansions is

Example:

The coefficients gere have a special signifucance.isthe number of ways in which a term with canbe generated whenisexpanded fully.

Suppose we have 5 men of which three are to be selected forpromotion. The number of ways in which three men can be selected fromthe five is the samegeneratedby the binomial expansion above. When choosing a selection offroma possiblewesay 'choose'We may write it asor

Given a set ofindividuals,the total number of selections of any size that can be made isThiscan also be deduced from the binomial theorem.

The total number of selections of size 0 is

The total number of selections of size 1 is

The total number of selections of size 2 is

The total number of selections of sizeis

The total number of selections of size n is

Adding all these up givesandwe can write this as