Find the number of integer solutions for $x_1+x_2+x_3 = 15$ under some constraints by IEP. For equation
$$
x_1+x_2+x_3 = 15
$$
Find number of positive integer solutions on conditions:
$$
x_1<6, x_2 > 6
$$
Let: $y_1 = x_1, y_2 = x_2 - 6, y_3 = x_3$
than, to solve the problem, equation $y_1+y_2 +y_3 = 9$ where $y_1 < 6,0<y_2, 0<y_3 $ has to be solved. Is this correct?
To solve this equation by inclusion-exclusion, number of solution without restriction have to be found $C_1 (3+9-1,9)$ and this value should be subtracted by $C_2 (3+9-7-1,2)$ , (as the negation of $y_1 < 6$ is $y_1 \geq 7$).
Thus:
$$
55-6=49
$$
Is this the correct answer ? 
Problem must be solved using inclusion-exclusion...

Here is a different way to break it down
$$
x_1\in\{1,2,3,4,5\}
$$
and given $x_1$ we then have $x_1+x_2<15$ and $x_2>6$ combined as
$$
6<x_2<15-x_1
$$
And whenever $x_1$ and $x_2$ are given, the value of $x_3$ follows from them.
For $x_1=5$ we then have $x_2\in\{7,8,9\}$ so three choices for $x_2$. Each time $x_1$ is decreased by $1$ we gain one option for $x_2$. Thus we have a total of
$$
3+4+5+6+7 = 25
$$
sets of integer solutions under the given constraints.

I ran the following code snippet in Python which confirmed the figure of 25:
n = 0
for x1 in range(1,16):
    for x2 in range(1,16):
        for x3 in range(1,16):
            if x1 < 6 and x2 > 6 and x1+x2+x3 == 15:
                n += 1
                print n, ":", x1, x2, x3

I understand that I did not answer the question using the method required, but I wonder why I find the number of solutions to be $25$ whereas the OP and the other answer find it to be $49$. Did I misunderstand the question in the first place?
