The Binomial Theorem has applications in many areas of mathematics, from calculus, to number theory, to probability. In this section we look at the connection between Pascal’s triangle and binomial coefficients. We ultimately prove the Binomial Theorem using induction.
Let be nonnegative integers with . An -combination of a set of elements is the number of subsets of size that can be chosen from a set of elements. We will use the notation .
Start with the right-hand side. Use the definition of “choose,” then find a common denominator. Note, you will save yourself a lot of work if you find the least common denominator.
We get the numbers in each row by adding the two numbers above. If there is only one number, you just get 1. For example, the fourth row is 1, 3, 3, 1, since . The next row would be 1, 4, 6, 4, 1.
If you calculate the binomial coefficients, you will see that you get the same values as Pascal’s triangle. Furthermore, Pascal’s formula, Theorem 6.1.4, is just the rule we use to get the triangle: add the and terms from the row to get the term in the row.
Why do we call a binomial coefficient? First, a binomial is an expression of the form . We will see that the Binomial Theorem gives a formula for calculating . The coefficients in this formula are the binomial coefficients.
We can generalize the counting argument from Example 6.1.5 to prove the Binomial Theorem. This is the type of proof you would encounter in a course which emphasizes counting techniques. However, we can also prove the Binomial Theorem using induction, which is more appropriate for this course.
Now we want to change the index of the second sum. This is just a substitution of variable that allows us to shift how we index the terms. If we were to write out the sum, rather than have it in summation notation, we would not need this step. But it allows us to easily combine like terms in the two summations. So, in the second sum, let , so when ; when , and . We get
where we pulled out the first term of the first sumand the last term of the second sumwhere we just relabeled the index in the second sumwhere we combined like terms in the two sumsby Pascal's Formula