<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] Following answer is for general case.
Denote by $c_m(1,n)=\binom{n-1}{m-1}$ the number of composition of $n$ into $m$ positive parts, and by $c_m(2,n)$ number of compositions of $n$ into $m$ parts greater than $1$. 
Each composition of $n$ into $m$ parts greater than $1$, if all parts decrease by 1, becomes a composition of $n-m$ into $m$ positive parts. That means $$c_m(2,n)=c_m(1,n-m)=\binom{n-m-1}{m-1}$$
Number of all compositions of $n$ into parts greater than $1$ is $$c(2,n)=\sum_{m=1}^{n}c_m(2,n)=\sum_{m=1}^{n}\binom{n-m-1}{m-1}$$
In our case $n=12$
$$c(2,12)=\sum_{m=1}^{12}\binom{11-m}{m-1}=\sum_{m=1}^{6}\binom{11-m}{m-1}=$$
$$=\binom{10}{0}+\binom{9}{1}+\binom{8}{2}+\binom{7}{3}+\binom{6}{4}+\binom{5}{5}=$$
$$1+9+28+35+15+1=89$$ 
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