## Task 4 - 280834

For a school sports festival, class $8 c$ wants to select a team of four students from the seven best students in the 100-m sprint to participate in the $4 \times 100$-m relay race.

a) How many different teams could be selected from the seven students?

b) How many different teams could be selected from the students if two specific students out of the seven must be included?

c) How many different teams could be selected from the students if three specific students out of the seven must be included?

d) In how many different orders of their starts can the four students of a team always be arranged for the relay race?
a) To initially select one of the seven students, there are exactly 7 possibilities.

To then select a second student, for each of the 7 mentioned possibilities, there are exactly 6 continuation possibilities. Of the thus found $7 \cdot 6$ possibilities, exactly 2 lead to the same selection of two students.

Therefore, there are exactly $\frac{7 \cdot 6}{2}=21$ possibilities for selecting two of the seven students.

This process can be continued. To select a third student, for each of the 21 possibilities, there are exactly 5 continuation possibilities. Of the thus found 21 . 5 possibilities, exactly 3 lead to the same selection; therefore, there are exactly 35 possibilities for selecting three of the seven students.

Correspondingly, it is found that there are 35 possibilities for selecting four of the seven students.
b) If two specific students out of the seven must be included, one only needs to select two of the remaining five students. As in a), there are exactly 10 possibilities for this selection.

c) If three specific students out of the seven must be included, one only needs to select one of the remaining four students. For this selection, there are exactly 4 possibilities.

d) To initially select the first to start among the four students, there are exactly 4 possibilities. In each of these possibilities, there are exactly 3 possibilities for selecting the second to start. In each of the resulting 12 possibilities, there are exactly 2 possibilities for the third; the fourth is then fixed. Therefore, there are exactly 24 possibilities for the starting order of a team.