Hit & Trial?

Consider cases when $z = 0,1,2, \ldots , 11$
$\Rightarrow x + y = 33,30,27, \ldots$

Total no. of solutions when $x + y = 33,30,27 \ldots$ is $^{33+2-1}C_{2-1} , ^{30+2-1}C_{2-1} , ^{27+2-1}C_{2-1}, \ldots$

So, Total no. of solutions = $34+31+28+ \ldots + 1 \quad (\text{12 terms}) = \dfrac{12}{2}(1+34) = \large {\color{#3D99F6}{210}}$

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Details :

• No. of non-negative integral solution of $x_1 + x_2 + \ldots x_n = k$ is $^{n+k-1}C_{n-1}$

• $^nC_r = \dbinom{n}{r}$