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## 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 [tex] x_1 + x_2 +x_3+x_4=25, x_1>0, x_2>0, x_3\le5, x_4\ge4 [/tex]

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 [tex] x_1 + x_2 + x_3 \ldots x_n = C, x_i>0 [/tex] is [tex] C-1\choose n-1 [/tex].

## The Attempt at a Solution

EDIT: got the solution

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