How many triples (a,b,c) of even positive integers satisfy ?

Note: this is solution is incorrect, since I missed the less than, I thought it was just an equal sign.

We first need to determine the range of for $a$:

$1^3=1,\\ 2^3=8,\\ 3^3=27,\\ 4^3=64.\\ \Rightarrow{a=2}$

$a=2, \text{since a is even}\\ \Rightarrow2^3+b^2+c=50\\ b^2+c=42$

Repeating our process, b can be 2, 4, or 6

There is a fixed value for c for each value of b. 

Therefore, there are 3 triples that satisfy $a^3+b^2+c=50$

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How many triples (a,b,c) of even positive integers satisfy

$a^3+b^2+c\le 50$

a^3+b^2+c\le 50 ?

$\begin{array}{|r|r|r|r|r|} \hline & a & b & c & a^3+b^2+c \le 50 \\ \hline 1 & 2&2&2 & 14 \\ 2 & 2&2&4 & 16 \\ 3 & 2&2&6 & 18 \\ 4 & 2&2&8 & 20 \\ 5 & 2&2&10 & 22 \\ 6 & 2&2&12 & 24 \\ 7 & 2&2&14 & 26 \\ 8 & 2&2&16 & 28 \\ 9 & 2&2&18 & 30 \\ 10 & 2&2&20 & 32 \\ 11 & 2&2&22 & 34 \\ 12 & 2&2&24 & 36 \\ 13 & 2&2&26 & 38 \\ 14 & 2&2&28 & 40 \\ 15 & 2&2&30 & 42 \\ 16 & 2&2&32 & 44 \\ 17 & 2&2&34 & 46 \\ 18 & 2&2&36 & 48 \\ 19 & 2&2&38 & 50 \\ \hline 20 & 2& 4& 2 & 26 \\ 21 & 2& 4& 4 & 28 \\ 22 & 2& 4& 6 & 30 \\ 23 & 2& 4& 8 & 32 \\ 24 & 2& 4& 10 & 34 \\ 25 & 2& 4& 12 & 36 \\ 26 & 2& 4& 14 & 38 \\ 27 & 2& 4& 16 & 40 \\ 28 & 2& 4& 18 & 42 \\ 29 & 2& 4& 20 & 44 \\ 30 & 2& 4& 22 & 46 \\ 31 & 2& 4& 24 & 48 \\ 32 & 2& 4& 26 & 50 \\ \hline 33 & 2& 6& 2 & 46 \\ 34 & 2& 6& 4 & 48 \\ 35 & 2& 6& 6 & 50 \\ \hline \end{array}$

$\text{ $\mathbf{35}$ triples $(a,b,c)$ of even positive integers satisfy $a^3+b^2+c \le 50$ }$