$\begin{align*} &\left [ {\color{red}{\bf x^{83}}} \right ] : \left [ \left ( x^5+x^8 + x^{11} + x^{14} + x^{17}\right )^{10} \right ] \\ &\left [ {\color{red}{\bf x^{83}}} \right ] :\left [ x^{50}\left ( 1+x^3 + x^{6} + x^{9} + x^{12} \right )^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] :\left [\left ( 1+x^3 + x^{6} + x^{9} + x^{12} \right )^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] :\left [ \left \{ 1+(x^3)^1 + (x^{3})^2 + (x^{3})^3 + (x^{3})^4 \right \}^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \left \{ \frac{1-(x^3)^{4+1}}{1-(x^3)} \right \}^{10} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \left ( 1-x^{15} \right )^{10}.\frac{1}{\left ( 1-x^3 \right )^{10}} \right ] \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \sum_{r=0}^{10}\binom{10}{r}\left ( -x^{15} \right )^r \; . \; \sum_{k=0}^{\infty}\binom{10+k-1}{k} . (x^3)^k \right ] \\ \end{align*}$
Now,
$\begin{align*} &\left [ {\color{red}{\bf x^{33}}} \right ] : \left [ \sum_{r=0}^{10}\binom{10}{r}\left ( -x^{15} \right )^r \; . \; \sum_{k=0}^{\infty}\binom{10+k-1}{k} . (x^3)^k \right ] \\ \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \begin{cases} (-1)^0.\binom{10}{0}.\binom{10+11-1}{11} \qquad r = 0, k = 11 \\ \\ (-1)^1.\binom{10}{1}.\binom{10+6-1}{6} \qquad r = 1, k = 6 \\ \\ (-1)^2.\binom{10}{2}.\binom{10+1-1}{1} \qquad r = 2, k = 1 \\ \end{cases} \\ \\ &\left [ {\color{red}{\bf x^{33}}} \right ] : \begin{cases} +\binom{10}{0}.\binom{20}{11} \qquad r = 0, k = 11 \\ \\ -\binom{10}{1}.\binom{15}{6} \qquad r = 1, k = 6 \\ \\ +\binom{10}{2}.\binom{10}{1} \qquad r = 2, k = 1 \\ \end{cases} \end{align*}$
NOTE:
1. $1+x+x^2+x^3+.....x^n = \frac{1-x^{n+1}}{1-x}$
2. $\frac{1}{(1-x)^n} = \sum_{r=0}^{\infty}\binom{n+r-1}{r}.x^r$
3. $\left [ x^{83} \right ]$ means coefficient of $x^{83}$ of the whole expression.