How Many Ways to Distribute Pencils with Constraints?

[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 $$x_1 + x_2 +x_3+x_4=25, x_1>0, x_2>0, x_3\le5, x_4\ge4$$

I think the end result was like 132 or something.

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.

Edit: accidently hit some random buttons and it submitted wrong, the please see my other forum post titled "Elementary Combinatorics Q"

 

[lippyka]

Homework EquationsThe attempt at a solutionI used stars and bars to solve this problem. First, set up the stars and bars equation: x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4=25For the stars and bars equation, you need an equation for each object (x's) and another for the gaps between them (y's). For the x's, since Ahmed and Dieter must receive at least one pencil, the equation is: x_1 + x_2 = 25-x_3-x_4-y_1-y_2-y_3-y_4For Carlos, since he can't receive more than 5 pencils, the equation is: x_3 \le 5For Barbara, since she must receive at least 4 pencils, the equation is: x_4 \ge 4 Since there are no restrictions on the y's, the equation is just: y_1 + y_2 + y_3 + y_4=0 Then I solved the stars and bars equation and got 132 solutions.