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 find the coefficient of the third term in the expansion of \((a+b)^6\), we can use Pascal's triangle, which gives the coefficients for the binomial expansion.

The \(n\)-th row of Pascal's triangle corresponds to the coefficients of \((a + b)^n\). In this case, \(n = 6\).

The 6th row of Pascal's triangle is as follows:

\[
1, 6, 15, 20, 15, 6, 1
\]

The terms of the expansion of \((a + b)^6\) can be represented as follows:

\[
\text{Coefficient} \cdot a^{6-k} \cdot b^k
\]

where \(k\) is the index of the term starting from \(0\). Therefore, the \(k\)-th term corresponds to the coefficient in position \(k\) of the 6th row.

- The **first term** corresponds to \(k = 0\): coefficient is \(1\).
- The **second term** corresponds to \(k = 1\): coefficient is \(6\).
- The **third term** corresponds to \(k = 2\): coefficient is \(15\).

Thus, the coefficient of the third term in the expansion of \((a + b)^6\) is \(\boxed{15}\).