Problem 29 - Beginning and Intermediate Algebra

Question

How many terms are in the expansion of \((a+b)^{9} ?\)

Step-by-Step Solution

Answer

There are 10 terms in the expansion of \((a+b)^{9}\).

Step 1: Recall the binomial theorem

The binomial theorem states that for any natural number n, the expansion of \((a+b)^n\) can be written as a sum of terms, each of which has a "binomial coefficient" multiplied by a power of a and a power of b. In general, the binomial theorem can be written as: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\), where \(\binom{n}{k}\) denotes the binomial coefficient, which is defined as: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where n! denotes the factorial of n (i.e., the product of all positive integers up to and including n).

Step 2: Determine the number of terms using the binomial theorem

In our case, we have \(n = 9\), so the expansion of \((a+b)^9\) will have a sum of terms from \(k = 0\) to \(k = n\). This means that the expansion will have \(n + 1\) terms, because we start counting from \(k = 0\). So, the number of terms in the expansion of \((a+b)^9\) will be: \(9 + 1 = 10\).

Step 3: Final answer

There are 10 terms in the expansion of \((a+b)^{9}\).