Find the total number of coefficients in the expansion of $(x^2-x +1)^{22}$ that are divisible by $6$. Find the total number of coefficients in the expansion of $(x^2-x +1)^{22}$ that are divisible by $6$.
So I am really stuck on this one.. I immediately defeated when I saw that I need to compute  $\binom {22} {a,b,c}$ where $a,b,c$ is the coefficients of $x^2,-x,1$. So what should be the next step I need to take? 

You can find which coefficients are divisible by $3$ by noting that $x^2-x+1\equiv(x+1)^2\pmod{3}$ so we are seeking the coefficients of $(x+1)^{44}$ which are divisible by $3$. Now notice that:
$$(x+1)^{44}=(x+1)^{27}(x+1)^{9}(x+1)^{6}(x+1)^2\equiv (x^{27}+1)(x^9+1)(x^{6}-x^3+1)(x^2-x+1)$$
Inspection sees the zero terms are $x^{18},x^{19},x^{20},x^{24},x^{25},x^{26}$.
Of these, you need to figure out which are even. That part is a little harder.
