Đáp án: 1695
Giải thích các bước giải:
$\begin{array}{l}
{\left( {1 + x + 3{x^2}} \right)^{10}}\\
= \sum\limits_{k = 0}^{10} {C_{10}^k.{{\left( {3{x^2}} \right)}^{10 - k}}.{{\left( {1 + x} \right)}^k}} \\
= \sum\limits_{k = 0}^{10} {\sum\limits_{m = 0}^k {C_{10}^k.C_k^m{{.3}^{10 - k}}.{x^{20 - 2k + m}}} } \\
\Rightarrow 20 - 2k + m = 4\\
\Rightarrow 2k - m = 16\left( {k \le 10;m \le k} \right)\\
\Rightarrow \left[ \begin{array}{l}
k = 10;m = 4\\
k = 9;m = 2\\
k = 8;m = 0
\end{array} \right.\\
\Rightarrow hệ\,số\,{x^4}:C_{10}^{10}.C_{10}^4{.3^{10 - 10}} + C_{10}^9.C_9^2{.3^1} + C_{10}^8.C_8^0{.3^2} = 1695
\end{array}$
