We explore the Binomial Theorem.
- Let and consider the coefficient of the term in the expansion of . According to the distributive property, both terms in each factor multiply both of the terms in each of the other factors. Hence, the coefficient sought is the number of ways to select of the (and hence simultaneously of the ) from the factors. Since the order of the selection is not important, we seek the number of combinations of objects, taken at a time, i.e., .
Pascal’s Triangle contains the binomial coefficients arranged in a triangular array.
From Pascal’s Identity, each number in the interior of the triangle is the sum of the two numbers above it. Try to write the next 3 rows of Pascal’s Triangle for yourself. Pascal’s Triangle can be used to expand a binomial expression.
The term involving will have the form Thus, the coefficient of is
b) Find the coefficient of in the expansion of
c) Find the coefficient of in the expansion of
If we substitute and , the Binomial Theorem gives, which simplifies to the identity in question.
If we substitute and , the Binomial Theorem gives, Rewriting as we arrive at
and it’s derivative, Evaluating we have Hence,
We close this section by examining the function . According to the Binomial Theorem, Observe that the coefficient of in is the binomial coefficient . Polynomial functions (and power series functions) whose coefficients contain combinatorial information are known as generating functions. We will study generating functions in chapter 4.