By Winter P. A.

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**Example text**

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48 3. 2. How to distribute n pennies to k children? 4 Distributing Money Instead of distributing presents, let’s distribute money. Let us formulate the question in general: We have n pennies that we want to distribute among k kids. Each child must get at least one penny (and, of course, an integer number of pennies). How many ways can we distribute the money? Before answering this question, we must clarify the diﬀerence between distributing money and distributing presents. If you are distributing presents, you have to decide not only how many presents each child gets, but also which of the diﬀerent presents the child gets.

First, what are the row sums? We get 1 for Al and 0 for everybody else. This is not a coincidence. If we consider a student like Al, who does not have any picture, then this student contributes to the bonus column, but nowhere else, which means that the sum in the row of this student is 1. Next, consider Ed, who has all 3 pictures. He has a 1 in the bonus column; in the next 3 columns he has 3 terms that are −1. In each of the next 3 columns he has a 1, one for each pair of pictures; it is better to think of this 3 as 32 .

### 2-3 graphs which have Vizings adjacency property by Winter P. A.

by Charles

4.3