回復 1# martinofncku 的帖子
\(\begin{align}
& \left( 1+x+{{x}^{2}}+\cdots +{{x}^{9}} \right){{\left( x+{{x}^{2}}+{{x}^{3}}+\cdots +{{x}^{9}} \right)}^{3}} \\
& ={{x}^{3}}\left( 1+x+{{x}^{2}}+\cdots +{{x}^{9}} \right){{\left( 1+x+{{x}^{2}}+\cdots +{{x}^{8}} \right)}^{3}} \\
\end{align}\)
原題即求\(\left( 1+x+{{x}^{2}}+\cdots +{{x}^{9}} \right){{\left( 1+x+{{x}^{2}}+\cdots +{{x}^{8}} \right)}^{3}}\)之\({{x}^{9}}\)的係數
轉化成以下整數解之組數
\(\begin{align}
& 0\le a\le 9 \\
& 0\le b\le 8 \\
& 0\le c\le 8 \\
& 0\le d\le 8 \\
& a+b+c+d=9 \\
\end{align}\)
所求\(=H\left( 4,9 \right)-3=217\)
