<s>[INST] In how many ways i can write 12? In how many ways i can write 12 as an ordered sum of integers where
the smallest of that integers is 2? for example 2+10 ; 10+2 ; 2+5+2+3 ; 5+2+2+3;
2+2+2+2+2+2;2+4+6; and many more
 [/INST] You are looking for number of partitions of 12 in parts greater than 1. Because their number is not so big we can present all of them in a list bellow
$$\begin{array}{c|c}
2+2+2+2+2+2 & 1 \\
2+2+2+2+4 &  5 \\
2+2+2+3+3 &  10 \\
2+2+2+6 &  4 \\
2+2+3+5 & 12 \\
2+2+4+4 & 6 \\
2+3+3+4 & 12 \\
3+3+3+3 & 1 \\
2+2+8 & 3 \\
2+3+7 & 6 \\
2+4+6 & 6 \\
2+5+5 & 3 \\
3+3+6 & 3 \\
3+4+5 & 6 \\
4+4+4 & 1 \\
2+10 & 2 \\
3+9 & 2 \\
4+8 & 2 \\
5+7 & 2 \\
6+6 & 1 \\
12 & 1 \\
\end{array}$$
The partition turn in composition if we rearrange them. Number of arrangements is given on the right, so total number of compositions is $89$
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