Question

Using Pascal’s Triangle, what is the coefficient of the third term in the expansion of (a+b)^6 when the expanded polynomial is written in standard form?

Answer 1

To determine the coefficient of the third term in the expansion of \((a + b)^6\), we will refer to Pascal's Triangle.

In general, the Binomial Theorem states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

Here, \(n = 6\). The third term corresponds to \(k = 2\) (because the term indexing starts from \(k = 0\)). Therefore, we need to compute:
\[
\binom{6}{2} a^{6-2} b^2 = \binom{6}{2} a^{4} b^{2}
\]

Next, we calculate the binomial coefficient \(\binom{6}{2}\):
\[
\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
\]

Thus, the third term in the expansion of \((a + b)^6\) is:
\[
15 a^{4} b^{2}
\]

Consequently, the coefficient of the third term is:
\[
\boxed{15}
\]