Counting Possible Distributions of Identical Pencils Among Four Students

[blinktx411]

Homework Statement

Suppose that a teacher wishes to distribute 25 identical pencils to Ahmed, Bar-
bara, Carlos, and Dieter such that Ahmed and Dieter receive at least one pencil
each, Carlos receives no more than five pencils, and Barbara receives at least four
pencils. In how many ways can such a distribution be made?

Or, in other words, find integer solutions to $$x_1 + x_2 +x_3+x_4=25, x_1>0, x_2>0, x_3\le5, x_4\ge4$$

Please let me know if i made any silly errors, but I'm more concerned that I made a fundamental error in the logic of this problem. Thanks!
The first inequality is

Homework Equations

The number of integer solutions to the equation $$x_1 + x_2 + x_3 \ldots x_n = C, x_i>0$$ is $$C-1\choose n-1$$.

The Attempt at a Solution

EDIT: got the solution

 

[Random Variable]

I would use generating functions.

Expand (x+x^2+...+x^25)*(x+x^2+...+x^25)*(1+x+x^2+x^3+x^4+x^5)*(x^4+x^5+...x^25) and find the coefficient of x^25.

I'll do it on Maple and see what I get.

 

[Random Variable]

I'm getting 980.

 

[blinktx411]

looking it over again my attempt at a solution is all backwards =/

 

[blinktx411]

Tada! I'm getting 980 after looking it over as well, don't know what I was thinking when I first attempted the solution. Thanks for your responses, I'll post my solution when I have more time.