Number of garlands which could be formed by using $3$ flowers of same type and $12$ flowers of other type. Question : Find number of garlands which could be formed by using $3$ flowers of same type and $12$ flowers of other type.

By garland I mean a perfectly circular one .

All $3$ flowers are alike of one type, all other $12$ are alike of different type.

An attempt at the solution.

Firstly we arrange all the $3$ of one type flowers then arrange the other ones, now after arranging the $3$ my circle got divided into $3$ parts and the sum of the flowers in that part has to be $12$.
Let the three parts be $a, b$, and $c$, so
$$a+b+c=12$$
Now the number of integral solutions (without considering the order as flowers are alike) of this equation will be my answer it should be $14\choose 2$.
However, it is wrong. Somebody help me out.

The numbers that you are after are called compositions (with 0s allowed, rotated such that the largest number comes first, and are not of the form a + b + a with a and b distinct). The compositions of 12 into 3 parts are:
12 + 0 + 0
11 + 1 + 0
11 + 0 + 1
10 + 2 + 0
10 + 1 + 1
10 + 0 + 2
9 + 3 + 0
9 + 2 + 1
9 + 1 + 2
9 + 0 + 3
8 + 4 + 0
8 + 3 + 1
8 + 2 + 2
8 + 1 + 3
8 + 0 + 4
7 + 5 + 0
7 + 4 + 1
7 + 3 + 2
7 + 2 + 3
7 + 1 + 4
7 + 0 + 5
6 + 6 + 0
6 + 5 + 1
6 + 4 + 2
6 + 3 + 3
6 + 2 + 4
6 + 1 + 5
5 + 5 + 2
5 + 4 + 3
5 + 3 + 4
4 + 4 + 4  
So there are 31 of these. 
