3.1.3 Sum and Product of Coefficients in Binomial Expansion

From the binomial expansion we know that:

$$(ax + by)^n = \sum_{k=0}^{n} \binom{n}{k} (ax)^{n-k} (by)^k$$ $$= \binom{n}{0} a^n \cdot x^n + \binom{n}{1} a^{n-1}b^1 \cdot x^{n-1}y^1 + \dots + \binom{n}{n} b^n y^n$$

From here we can see that the sum of the coefficients of the expansion is:

$$\sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$$

Where we can retrieve these sums by setting $x = 1$ and $y = 1 \Rightarrow$ Sum of the Coefficients = $(a + b)^n$.

Here is an example to clear things up:

Problem: Find the Sum of the Coefficients of $(x + y)^6$. Solution: Let $x = 1$ and $y = 1$ which leads to the Sum of the Coefficients = $(1 + 1)^6 = 64$.

An interesting side note on this is when asked to find the Sum of the Coefficients of $(x - y)^n$ it will always be 0 because by letting $x = 1$ and $y = 1$ you get the Sum of the Coefficients = $(1 - 1)^n = 0$.

As for the product of the coefficients, there are no easy way to compute them. The best method is to memorize some of the first entries of the Pascal triangle (if you’re unfamiliar with how Pascal’s triangle relates to the coefficients of expansion, I suggest Googling it):

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1
 1 5 10 10 5 1
1 6 15 20 15 6 1

Here are some more practice to get acquainted with both the sum and product of coefficients:

Problem Set 3.1.3