Expand (x1 + 2x2 + 3x3)4.

Step 1 of 2

Our goal is:

We need to expand \(\left(x_{1}+2 x_{2}+3 x_{3}\right)^{4}\).

The multinomial formula is

\(\begin{array}{l}
\left(x_{1}+x_{2}+x_{3}, \ldots,+x_{r}\right)^{n}=\sum_{\left(n_{1}, n_{2}, \ldots, n_{r}\right)}\left(n_{1}^{n}, n_{2}, \ldots, n_{r}\right) x_{1}^{n_{1}} x_{2}^{n_{2}, \ldots, x_{r}^{n_{r}}} \\
\left(x_{1}+x_{2}+x_{3}, \ldots,+x_{r}\right)^{n}=\sum_{n_{1}+n_{2}+, \ldots,+n_{r}=n}\left(n_{1}^{n}, n_{2}, \ldots, n_{r}\right) x_{1}^{n_{1}} x_{2}^{n_{2}}, \ldots, x_{r}^{n_{r}}
\end{array}\)

The sum is overall nonnegative integer-valued vectors \(\left(n_{1}+n_{2}+, \ldots,+n_{r}\right)\) is

\(\left(n_{1}+n_{2}+, \ldots,+n_{r}\right)=n\).

Given, \(\left(x_{1}+2 x_{2}+3 x_{3}\right)^{4} \text {. }\)

\(\begin{array}{l}
\left(x_{1}+2 x_{2}+3 x_{3}\right)^{4}=\sum_{n_{1}+n_{2}+n_{3}=4}\left(n_{1}^{4}, n_{2}, n_{3}\right) x_{1}^{n_{1}}\left(2 x_{2}\right)^{n_{2}}\left(3 x_{3}\right)^{n_{3}} \\
\left(x_{1}+2 x_{2}+3 x_{3}\right)^{4}=\left(\begin{array}{c}
4 \\
4,0,0
\end{array}\right) x_{1}^{4}+\left(\begin{array}{c}
4 \\
3,1,0
\end{array}\right) x_{1}^{3}\left(2 x_{2}\right)^{1}+\left(\begin{array}{c}
4 \\
3,0,1
\end{array}\right) x_{1}^{3}\left(3 x_{3}\right)^{1}+\left(\begin{array}{c}
4 \\
2,2,0
\end{array}\right) x_{1}^{2}\left(2 x_{2}\right)^{2}
\end{array}\)                                             

\(\begin{array}{l}
+\left(\begin{array}{c}
4 \\
2,1,1
\end{array}\right) x_{1}^{2}\left(2 x_{2}\right)^{1}\left(3 x_{3}\right)^{1}+\left(\begin{array}{c}
4 \\
2,0,2
\end{array}\right) x_{1}^{2}\left(3 x_{3}\right)^{2}+\left(\begin{array}{c}
4 \\
1,3,0
\end{array}\right) x_{1}^{1}\left(2 x_{2}\right)^{3} \\
+\left(\begin{array}{c}
4 \\
1,2,1
\end{array}\right) x_{1}^{2}\left(2 x_{2}\right)^{1}\left(3 x_{3}\right)^{1}+\left(\begin{array}{c}
4 \\
1,1,2
\end{array}\right) x_{1}^{1}\left(2 x_{2}\right)^{1}\left(3 x_{3}\right)^{2}+\left(\begin{array}{c}
4 \\
1,0,3
\end{array}\right) x_{1}^{1}\left(3 x_{3}\right)^{3} \\
+\left(\begin{array}{c}
4 \\
0,4,0
\end{array}\right)\left(2 x_{2}\right)^{4}+\left(\begin{array}{c}
4 \\
0,3,1
\end{array}\right)\left(2 x_{2}\right)^{3}\left(3 x_{3}\right)^{1}+\left(\begin{array}{c}
4 \\
0,2,2
\end{array}\right)\left(2 x_{2}\right)^{2}\left(3 x_{3}\right)^{2} \\
+\left(\begin{array}{c}
4 \\
0,1,3
\end{array}\right)\left(2 x_{2}\right)\left(3 x_{3}\right)^{1}+\left(\begin{array}{c}
4 \\
0,0,4
\end{array}\right)\left(3 x_{3}\right)^{4}
\end{array}\)