5. At a birthday party, 5 girls and 5 boys gathered. They were playing a game where two teams compete, with each team consisting of 2 boys and 1 girl. In how many ways could the teams be formed for the game, ensuring that the opponents are different?

Each task is scored out of 10 points.

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

Ministry of Science, Education and Sports of the Republic of Croatia

Agency for Education and Upbringing

Croatian Mathematical Society

## REGIONAL COMPETITION IN MATHEMATICS
5. For the 1st group, the 1st boy can be chosen in 5 ways, and after that, 4 boys remain, so the 2nd boy can be chosen in 4 ways.

Since the order in which the boys are chosen does not matter, the total number of different ways to choose 2 boys for the 1st group is (5-4):2=10.

For the 1st group, 1 girl can be chosen in 5 ways, so the total number of different ways to choose the 1st group is $10 \cdot 5=50$.

After choosing the 1st group, 3 boys and 4 girls remain for the 2nd group.

$1 \text{ POINT}$

For the 2nd group, 2 boys can be chosen in $(3 \cdot 2): 2=3$ ways, and 1 girl in 4 ways, so the total number of different ways to choose the 2nd group is $3 \cdot 4=12$.

Since it does not matter whether a person belongs to the 1st group or the 2nd group for the game, the total number of different ways to choose the groups for the game is $(50 \cdot 12): 2=300$. $\qquad$

## SOLUTIONS FOR 6th GRADE

HERE IS ONE WAY TO SOLVE THE PROBLEM. IF THE STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE MEMBER OF THE COMMITTEE IS OBLIGED TO EVALUATE AND GRADE THAT APPROACH IN AN APPROPRIATE MANNER.