<s>[INST] Finding the no. of non-negative integral solutions to $x+y+2z=33$. 
Finding the no. of non negative integral solutions to $x+y+2z=33$.

I tried many pure combinatorial approaches (I don't like making individual cases, too long). But they went in vain. I was just pondering, when randomly, a thought came to my mind.

$$x+y+2z=33$$
$$x+2y+z=33$$
$$2x+y+z=33$$
All would have the same number of solutions (integral). So if we add them up, then also, the solutions would remain the same. 
$$4x+4y+4z=99$$
$$x+y+z=24.75$$
Taking the floor of the right side (random), we get $x+y+z=24$. 
Then, the number of solutions would be ${26\choose2}$. But it is incorrect. So, what can I employ.
 [/INST] Clearly the number of non-negative integral solutions to the equations $x + y + 2z = 33$ is same as the number of similar solutions to $i + j + 2k = 33$. This is obviously equal to the coefficient of $x^{33}$ in the expansion $$(1 + x + x^{2} + \cdots + x^{i} + \cdots)(1 + x + x^{2} + \cdots + x^{j} + \cdots)(1 + x^{2} + \cdots + x^{2k} + \cdots)$$ which is the same as the coefficient of $x^{33}$ in $$\frac{1}{(1 - x)^{2}}\cdot\frac{1}{1 - x^{2}} = (1 + 2x + 3x^{2} + 4x^{3} + \cdots)(1 + x^{2} + x^{4} + x^{6} + \cdots)$$ and this coefficient is given by $$2 + 4 + 6 + \cdots + 32 + 34 = ((2 + 34)/2) \cdot 17 = 306$$
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