# Elementary Combinatorics Q

1. May 15, 2009

1. The problem statement, all variables and given/known data
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
2. Relevant equations

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

3. The attempt at a solution

EDIT: got the solution

Last edited: May 16, 2009
2. May 15, 2009

### 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.

3. May 15, 2009

### Random Variable

I'm getting 980.

4. May 15, 2009