2. On the island, there live 7 natives who know mathematics and physics, 6 natives who know physics and chemistry, 3 natives who know chemistry and mathematics, and 4 natives who know physics and biology. In how many ways can a team of three people be formed who together know at least three subjects out of the four? The four subjects are: mathematics, physics, chemistry, and biology.

#
# Answer: 1080.

## Solution.

Let's choose three people. If this trio includes representatives of at least two different groups of aborigines, then together they know no fewer than three subjects.

If this trio includes representatives of only one group, then together they know only two subjects.

Therefore, we will find the number of all possible trios and subtract from it the number of trios from representatives of one group. In total, there are $7+6+3+4=20$.

The first person can be chosen in 20 ways, the second in 19 ways, and the third in 18 ways. In total, by the multiplication rule, $20 \times 19 \times 18$. However, the chosen aborigines can be enumerated differently. The first can be appointed in 3 ways, the second in 2 ways, and the third in 1 way. In total, 6 ways of enumeration.

By the multiplication rule, $3 \times 2 \times 1=6$. Therefore, due to the enumeration of aborigines, we get 6 identical teams. Thus, the total number of trios is $(20 \times 19 \times 18): 6=20 \times 19 \times 3=1140$.

The number of trios from representatives of the first group will be ( $7 \times 6 \times 5$ ) : $6=35$.

The number of trios from representatives of the second group will be ( $6 \times 5 \times 4$ ): $6=20$.

The number of trios from representatives of the third group will be ( $3 \times 2 \times 1$ ): $6=1$.

The number of trios from representatives of the fourth group will be ( $4 \times 3 \times 2$ ): $6=4$.

Then the number of teams we need will be $1140-(35+20+1+4)=1080$.

Criteria. If the solution is incorrect - 0 points.

Only the correct answer, without justification - 1 point.

Only the number of all possible trios from representatives of two groups is found - 3 points.

Only the number of all possible trios from representatives of one group is found - 3 points.

The solution is completed with a computational error - 5 points. If the solution is correct - 7 points.