The twelve students in an olympiad class went out to play soccer every day after their math class, forming two teams of 6 players each and playing against each other. Each day they formed two different teams from those formed on previous days. By the end of the year, they found that every group of 5 students had played together on the same team exactly once. How many different teams were formed throughout the year?
For every group of 5 students, there is a unique team formed that contains them. Therefore, we have $C_{12}^{5}=\frac{12.11 .10 .9 .8}{5!}=792$ teams for each 5 students chosen. On the other hand, in each team of 6 players, there are $C_{6}^{5}=6$ ways to choose five players, that is, there are 6 groups of 5 players who were on the same team in our first count. Therefore, the total number of teams formed is equal to $\frac{792}{6}=132$.