Consider the following expansions of the power into a sum of terms:
•
|
|
•
|
|
•
|
|
This expansion can be expressed more compactly using the Binomial Formula:
.
The binomial coefficient , also written or and pronounced n choose k, is the number of ways of choosing a subset of k objects from a group of n objects. Its value can be given more explicitly as
for .
As you may have noticed, the coefficients of the expansion of the power correspond directly to the numbers in the row of Pascal's Triangle:
•
|
|
•
|
|
•
|
and so on...
|
In other words, the number in the position of the row of Pascal's Triangle is . For example, the number in position 0 of row 1 is while the number in position 2 of row 5 is .
Also, due to the symmetry of Pascal's triangle, we can easily see that . Finally, looking back upon the the original way of constructing the triangle, in which the two numbers in the row above are added together to get the current value, we see that
for and .