When we expand a power of a binomial expression we get a polynomial which can be considered as a series. It is not an arithmetic or geometric one but there is definitely a pattern.



Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.

Now that we are familiar with combinations, we know that each term in Pascal's triangle corresponds to a value of nCr.

In comparing the two triangles above, we will be able to observe that
Recall Pascal's method for creating his triangle:

With this comparison, the method must also apply to combinations, giving Pascal's formula:

Proof:

This proof allows us to see that the values of nCr do follow the pattern that creates Pascal's triangle.

Example 1: Applying Pascal's Formula to Combinations
Rewrite the following as the sum of 2 other combinations

The Binomial Theorem

When we expand a power of a binomial expression, the result is a polynomial, which can be considered as a series. It is not an arithmetic or geometric series, but there is a visible pattern occurring. This pattern leads us to the Binomial Theorem.

The Binomial Theorem is the formula used to obtain the expansion of powers of a binomial and is as follows:

Before we move on to a few more examples, let us look at some of the properties of the binomial expansion (a + b)n.

• There are n + 1 terms
• 
• Progressing from the first term to the last, the exponent of a decreases by 1 from term to term while the exponent of b increases by 1. In addition, the sum of the exponents of a and b in each term is n.
• If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.

Example 2: Applying the Binomial Theorem
Use combinations to expand the following


Example 3: Homework Question - # 20a on page 29



When finding the general term we apply the formula:

References