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 \(\boxed{15}\).