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

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 $89$
