5. Ante, Bruno, Ciprijan, Davor, Emanuel, and Franko need to line up in a row.

a) In how many different ways can the boys line up if Bruno stands to the left of Emanuel?

b) In how many different ways can the boys line up if there is no one standing between Ciprijan and Davor?

The use of a pocket calculator or any reference materials is not allowed.

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

## COUNTY COMPETITION IN MATHEMATICS  March 24, 2022.
First method:

If we were to arrange the boys without any conditions, we would sequentially choose one of the 6 boys for the first position, then one of the remaining 5 boys for the second position, and so on until the end. This can be done in $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$ ways.

2 POINTS

Notice that among these ways, each arrangement where Bruno is to the left of Emanuel has a counterpart where Bruno is to the right of Emanuel. We form pairs by swapping the positions of Bruno and Emanuel, while all others retain their positions.

3 POINTS

Therefore, the desired number is $720 \div 2 = 360$.

1 POINT