## Task B-1.7.

Two Italians and three Englishmen are looking for accommodation in a hotel as loyal guests. The hotel has 8 rooms reserved for accommodating its loyal guests, of which 3 rooms have walls painted blue, 3 rooms are painted green, and the remaining 2 rooms are painted yellow. The Italians want to be in either a blue or a yellow room, while the Englishmen want to be in either a green or a yellow room. In how many ways can the two Italians and three Englishmen be assigned rooms so that each of them is alone in one room?
## Solution.

1. case:

The Italians are assigned two blue rooms.

This can be done in $3 \cdot 2=6$ ways. 1 point

From 3 green and 2 yellow rooms, the English can be assigned 3 rooms in $5 \cdot 4 \cdot 3=60$ ways. 1 point

This is a total of $6 \cdot 60=360$ ways.

2. case:

The Italians are assigned two yellow rooms.

This can be done in $2 \cdot 1=2$ ways.

1 point

The three green rooms can be assigned to the English in $3 \cdot 2 \cdot 1=6$ ways. 1 point

This is a total of $2 \cdot 6=12$ ways.

1 point

3. case:

The Italians are assigned one blue and one yellow room.

We need to choose which of the two Italians gets the blue room (2 possibilities) and which of the blue rooms they will get (3 possibilities). Then the other Italian is assigned one of the 2 yellow rooms (2 possibilities).

This can be done in $2 \cdot 3 \cdot 2=12$ ways.

From 3 green and 1 yellow room, the English can be assigned 3 rooms in $4 \cdot 3 \cdot 2=24$ ways.

This is a total of $12 \cdot 24=288$ ways.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

## 2nd grade - high school - B variant

## February 15, 2022.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE TASK, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH IN AN APPROPRIATE MANNER.