General terms and coefficient selection

\(T_{r+1}=\binom nr a^{n-r}b^r\). The first term corresponds to \(r=0\), so the seventh term uses \(r=6\).

For \((Ax^p+Bx^q)^n\), \(T_{r+1}=\binom nr A^{n-r}B^r x^{p(n-r)+qr}\).

To find a specified power, equate the exponent of \(x\) to that power and solve for \(r\). Check that \(r\) is an integer between \(0\) and \(n\).

A term independent of \(x\) has exponent zero. If the required value of \(r\) is not an allowed integer, the expansion has no such term.

The term in \(x^k\) includes \(x^k\); its coefficient does not. Simplify the numerical factor only after choosing the correct term.