Answer
The fifth term: $17,010A^6B^4$
Work Step by Step
The $k+1$th term of the expansion of $(x+y)^n$ is:
$$T_{k+1}=\binom{n}{k}x^{n-k}y^k,\text{ where }k=0,1,2,\dots,n.\tag1$$
Substitute $x=A$, $y=3B$, $n=10$ in Eq. $(1)$ so that we find the expression of $T_{k+1}$ in the expansion of $(A+3B)^{10}$:
$$\begin{align*}
T_{k+1}&=\binom{10}{k}A^{10-k}(3B)^k.
\end{align*}$$
As $T_{k+1}$ has to contain $A^6$, we determine $k$ using the exponent of $A$:
$$\begin{align*}
10-k&=6\\
k&=4.
\end{align*}$$
Therefore $T_{k+1}=T_{4+1}=T_5$. The $5$th term of the expansion contains $A^6$. The term is:
$$T_5=\binom{10}{4}A^6(3B)^4=210A^6(81B^4)=17,010A^6B^4.$$
