Number of integral solutions of an equation with constraints I am having some trouble with finding the number of integral solutions of an equation with a constraint given. I understand how go about solving without a constraint, but I don't know how to solve it when a constraint is given. Here's an example problem:

Find out how many solutions there are to the equation $y_1 + y_2 + y_3 + y_4 = 55$, satisfying the condition that each $y_i$ is a positive integer greater than 7?

I know if there wasn't a constraint, we'd just use the $\binom{r + n - 1}{r}$ formula and do $\binom{55+4-1}{55}$, but I have no idea what to do about the constraints. Any help?
 A: Since $y_k$ is a positive integer greater than $7$, then $z_k = y_k - 8$ is a non-negative integer.  If we make the substitution $z_k + 8$ for $y_k$, $1 \leq k \leq 4$, we obtain
\begin{align*}
y_1 + y_2 + y_3 + y_4 & = 55 \tag{1}\\
z_1 + 8 + z_2 + 8 + z_3 + 8 + z_4 + 8 & = 55\\
z_1 + z_2 + z_3 + z_4 & = 23 \tag{2}
\end{align*}
Equation 2 is an equation with the same number of solutions in the non-negative integers as equation 1 has in the positive integers larger than $7$.  A particular solution of equation 2 corresponds to the placement of three addition signs in a row of $23$ ones.  For instance, 
$$1 1 1 1 1 1 1 + 1 1 1 1 1 1 1 1 1 1 + + 1 1 1 1 1 1$$
corresponds to the solution $z_1 = 7$, $z_2 = 10$, $z_3 = 0$, and $z_4 = 6$.  The number of solutions of equation 2 is equal to the number of ways three addition signs can be placed in a row of $23$ ones, which is 
$$\binom{23 + 3}{3} = \binom{26}{3}$$
since we must select which three of the twenty-six symbols (twenty-three ones and three addition signs) are addition signs.    
A: Define $x_j = y_j - 7,$ and count the number of ways for the $x$s to add to $55-28 = 27$
Note: the other answers prefer subtracting $8$ so as to get $x_j \geq 0,$ sum $55-32 = 23.$ Well, why not?
A: A simple way is to pre-allot $8$ to each $y_i$,
so now the problem reduces to $y_1+y_2+y_3+y_4 = 23$, over non-negative integers
You already know what to do now.
