<s>[INST] A polynomial question. I came across this question in my weekly test paper and didn't know how to tackle it. The question goes as follows:
$a_0,a_1,a_2....a_{34}$ are the coefficients of $x^0,x^1,x^2......x^{34}$ of the polynomial obtained on opening the parenthesis of $(1+x+x^2)^{17}$, then which is true?
$1)~a_1+a_2+.......+ a_{34}$ is even;
$2)~a_1=17$;
$3)~a_{33}=17$;
$4)~a_2=153$.
(One or more is correct)
 [/INST] Here is some info regarding (2) - (4)

  
*
  
*Ad (2): $a_1$ is the coefficient of $x^1$ in $(1+x+x^2)^{17}$. Since there are $17$ factors of the form $1+x+x^2$ we have to choose precisely from one factor $x$ whereas from all other factors we have to choose $1$. This can be done in
  \begin{align*}
a_1=\binom{17}{1}=17
\end{align*}
  different ways.
  
*Ad (3): The polynomial $1+x+x^2=x\left(\frac{1}{x}+1+x\right)$ has a nice symmetrical structure. This implies the coefficient of $x^k$ in $(1+x+x^2)^{17}$ is also the coefficient of $x^{34-k}$ with $0\leq k \leq 34$. We conclude
  \begin{align*}
a_{33}=a_{34-1}=a_1=17
\end{align*}
  
*Ad (4): $a_2$ is the coefficient of $x^2$ in $(1+x+x^2)^{17}$. In order to get $x^2$ from $17$ factors of the form $1+x+x^2$ we have to choose one factor $x^2$ which can be done in $\binom{17}{1}$ different ways and $1$ from the remaining $16$ factors, or we have to choose two factors $x$ which can be done in $\binom{17}{2}$ different ways and $1$ from the remaining $15$ factors. We conclude
  \begin{align*}
a_2=\binom{17}{1}+\binom{17}{2}=17+\frac{1}{2}\cdot 17\cdot 16=153
\end{align*}

</s>