Multinomial expansion

Solution for those who don't know Multinomial Theorem,

${ \left( 1+{ x }+2{ x }^{ 3 } \right) }^{ 10 }=\sum _{ r=0 }^{ 10 }{ { \left( \begin{matrix} 10 \\ r \end{matrix} \right) }{ \left( { x }+2{ x }^{ 3 } \right) }^{ r } } \\ \\ { min }^{ m }\quad power\quad of\quad x\quad in\quad { \left( { x }+2{ x }^{ 3 } \right) }^{ r }=r\\ \\ { max }^{ m }\quad power\quad of\quad x\quad in\quad { \left( { x }+2{ x }^{ 3 } \right) }^{ r }=3r,so\\ \\ r\le 7\le 3r\quad \Rightarrow \quad 2<r\le 7\\ \\ passible\quad r\quad are\quad 3,4,5,6,7\\ \\ \& \quad { \left( { x }+2{ x }^{ 3 } \right) }^{ r }={ x }^{ r }{ \left( 1+2{ x }^{ 2 } \right) }^{ r }={ x }^{ r }\sum _{ n=0 }^{ r }{ \left( \begin{matrix} r \\ n \end{matrix} \right) { 2 }^{ n }{ x }^{ 2n } } \\ \\ 2n+r=7\quad so\quad r\quad only\quad odd\\ \\ so\quad now\quad only\quad passible\quad \left( r,n \right) \quad are\quad \left( 3,2 \right) ,\left( 5,1 \right) ,\left( 7,0 \right) \\ \\ \therefore coff.\quad of\quad { x }^{ 7 }\quad in\quad { \left( 1+{ x }+2{ x }^{ 3 } \right) }^{ 10 }\\ \\ \quad \quad \quad =coff.\quad of\quad { x }^{ 7 }\quad in\quad \left\{ { \left( \begin{matrix} 10 \\ 3 \end{matrix} \right) }{ \left( { x }+2{ x }^{ 3 } \right) }^{ 3 }{ +\left( \begin{matrix} 10 \\ 5 \end{matrix} \right) }{ \left( { x }+2{ x }^{ 3 } \right) }^{ 5 }+{ \left( \begin{matrix} 10 \\ 7 \end{matrix} \right) }{ \left( { x }+2{ x }^{ 3 } \right) }^{ 7 } \right\} \\ \\ \quad \quad \quad =\left\{ \left( \begin{matrix} 10 \\ 3 \end{matrix} \right) \left( \begin{matrix} 3 \\ 2 \end{matrix} \right) { 2 }^{ 2 }+\left( \begin{matrix} 10 \\ 5 \end{matrix} \right) \left( \begin{matrix} 5 \\ 1 \end{matrix} \right) { 2 }^{ 1 }+\left( \begin{matrix} 10 \\ 7 \end{matrix} \right) \left( \begin{matrix} 7 \\ 0 \end{matrix} \right) { 2 }^{ 0 } \right\} \\ \\ \quad \quad \quad =4080$

$(1 + x + 2x^3)^10 = [ ( 1 + x ) + 2x^3 ]^10$

You only need to consider these three terms:

(1 + x)^10

(1+x)^9 · 2x^3

(1 + x)^8 · (2x^3)^2

Other terms don't have x^7.

The coefficient of x^7 in (1 + x)^10 is 10C0 · 10C7 = 120

The coefficient of x^7 in (1+x)^9 · 2x^3 is 10C1 · 9C4 · 2 = 2520

The coefficient of x^7 in (1 + x)^8 · (2x^3)^2 is 10C2 · 8C1 · 2^2 = 1440

120 + 2520 + 1440 = 4080

$1 + 10 x + 45 x^2 + 140 x^3 + 390 x^4 + 972 x^5 + 2070 x^6 + 4080 x^7 + 7605 x^8 + 12730 x^9 + 20041 x^10 + 30420 x^11 + 42020 x^12 + 55200 x^13 + 70740 x^14 + 81984 x^15 + 91680 x^16 + 100800 x^17 + 97440 x^18 + 94080 x^19 + 88704 x^20 + 69120 x^21 + 59520 x^22 + 46080 x^23 + 26880 x^24 + 23040 x^25 + 11520 x^26 + 5120 x^27 + 5120 x^28 + 1024 x^30$