Đáp án:
\(C_7^3.C_3^0 + C_7^2.C_2^2 = 56\)
Giải thích các bước giải:
\(\begin{array}{l}
{\left[ {1 + {x^2}\left( {1 + x} \right)} \right]^7} = {\left( {{x^3} + {x^2} + 1} \right)^7}\\
= \sum\limits_{k = 0}^7 {C_7^k{{\left( {{x^3} + {x^2}} \right)}^k}} = \sum\limits_{k = 0}^7 {C_7^k\sum\limits_{i = 0}^k {C_k^i{x^{3i}}{x^{2\left( {k - i} \right)}}} } \\
= \sum\limits_{k = 0}^7 {\sum\limits_{i = 0}^k {C_7^kC_k^i{x^{2k + i}}} } \\
\Rightarrow De\,\,co\,\,he\,\,so\,\,cua\,\,{x^6} \Rightarrow \left\{ \begin{array}{l}
2k + i = 6\\
0 \le k \le 7\\
0 \le i \le k\\
i,\,\,k \in Z
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\left\{ \begin{array}{l}
i = 0\\
k = 3
\end{array} \right.\\
\left\{ \begin{array}{l}
i = 2\\
k = 2
\end{array} \right.
\end{array} \right.\\
\Rightarrow he\,\,so\,\,cua\,\,{x^6}\,\,trong\,\,khai\,\,trien\,\,tren\,\,la:\,\,\,C_7^3.C_3^0 + C_7^2.C_2^2 = 56.
\end{array}\)
