<s>[INST] How many ordered triples of integers which are between 0 and 10 inclusive do we actually have if $a * (b+c) = a * b +c$ How many ordered triples of integers $(a,b,c)$ which are between 0 and 10 inclusive do we have if:
 $a * (b+c) = a * b +c$
 [/INST] We have that
$$a(b+c)=ab+c \Rightarrow ab+ac=ab+c.$$
If $a=1$, then we have $b+c=b+c$, and if $c=0$ then $ab=ab$.
When $a=1$ there are $11 \cdot 11=121$ ways to write two of the eleven numbers from $0$ to $10$ inclusive for $b,c$ with repeat, and when $c=0$, there are again $11 \cdot 11=121$ ways to write two of the eleven numbers from $0$ to $10$ inclusive for $a,b$ with repeat. 
These two sets have eleven duplicates when $a=1$ and $c=0$, since $b$ is one of the same eleven values in both cases.
Hence there are $2 ( 11 \cdot 11)-11=231$ ordered triples $(a,b,c)$ that satisfy your equation.
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