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?(1 point)

Answer 1

To find the coefficient of the third term in the expansion of \((a + b)^6\) using Pascal's Triangle, we first need to recall that the \(n\)th row of Pascal's Triangle corresponds to the coefficients of the terms in the expansion of \((a+b)^n\).

For \(n = 6\), the coefficients in the expansion of \((a + b)^6\) can be found in the 6th row of Pascal's Triangle. The rows of Pascal's Triangle start from row 0.

The 6th row of Pascal's Triangle is:
\[
1, 6, 15, 20, 15, 6, 1
\]

The general term in the expansion of \((a+b)^n\) is given by the binomial coefficient \(\binom{n}{k} a^{n-k} b^k\), where \(k\) starts from 0.

In this case, the coefficients correspond to the following terms:
- For \(k = 0\) (first term): \( \binom{6}{0} a^6 b^0 = 1 a^6 \)
- For \(k = 1\) (second term): \( \binom{6}{1} a^5 b^1 = 6 a^5 b \)
- For \(k = 2\) (third term): \( \binom{6}{2} a^4 b^2 = 15 a^4 b^2 \)

Thus, the coefficient of the third term (where \(k = 2\)) is \(\binom{6}{2} = 15\).

Therefore, the coefficient of the third term in the expansion of \((a+b)^6\) is **15**.