5. In how many different ways can four children, Ante, Bruno, Cvijeta, and Dunja, distribute four identical pencils among themselves?

## Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society

## COUNTY COMPETITION IN MATHEMATICS March 24, 2022.
First solution.

Children share four identical pencils among themselves, and each of them can get a maximum of four pencils. However, it is possible that in some distribution, one of them does not get a single pencil. Therefore, we can have these five cases:

$$
\begin{aligned}
& 4=4+0+0+0 \\
& 4=3+1+0+0 \\
& 4=2+2+0+0 \\
& 4=2+1+1+0 \\
& 4=1+1+1+1
\end{aligned}
$$

Let's determine for each of these cases how many different ways the distribution of pencils can be made.

The distribution in which one child gets all four pencils can be made in 4 different ways. 1 POINT

The distribution in which one child gets three pencils and one child gets one pencil can be made in 12 different ways:

- the child who gets three pencils can be chosen in 4 ways
- for each choice of the child who gets three pencils, the child who will get one pencil can be chosen in 3 ways

2 POINTS

The distribution in which two children get two pencils each can be made in 6 different ways:

- the first child can be chosen in 4 ways, the second in 3 ways, but in this way, each selection is counted twice, so we divide 12 by 2

2 POINTS

The distribution in which one child gets two pencils and two children get one pencil each can be viewed as a distribution in which one child gets two pencils and one child gets none. Such a distribution can be made in 12 different ways:

- the selection of the one child who gets two pencils can be made in 4 different ways, and for each such selection, the child who will not get a pencil can be chosen in 3 ways (the remaining two children get one pencil each)

2 POINTS

The distribution in which each child gets one pencil can be made in 1 way.

1 POINT

Four children can share four identical pencils among themselves in

$4+12+6+12+1=35$ different ways.

1 POINT

TOTAL 10 POINTS

Note: If the student does not explicitly mention the five cases but examines and lists them, 1 POINT is awarded.