Combination problem

[BTGmoderatorRO]

Seven students are trying out for the school soccer team, on which there are three available positions: fullback, sweeper, and goalie. Each student can only try out for one position. The first two students are trying out for fullback. The next two students are trying out for sweeper. The remaining three students are trying out for goalie. However, the fourth student will only play if the second student is also on the team, and the third student will only play if the fifth student is on the team. How many possible combinations of students are there to fill the available positions?

A 3
B 5
C 7
D 10
E 12
OA is b

what is the best mathematical approach to use here? can any expert help?

Thanks for responding to my need

[GMATGuruNY]

Seven students are trying out for the school soccer team, on which there are three available positions: fullback, sweeper, and goalie. Each student can only try out for one position. The first two students are trying out for fullback. The next two students are trying out for sweeper. The remaining three students are trying out for goalie. However, the fourth student will only play if the second student is also on the team, and the third student will only play if the fifth student is on the team. How many possible combinations of students are there to fill the available positions?

A - 3
B - 5
C - 7
D - 10
E - 12

To keep track of the number of options for each position, draw a TREE.
Start with the MOST RESTRICTED position, which is SWEEPER.
If Sweeper = 3, then Goalie = 5.
If Sweeper = 4, then Fullback = 2.
Here's the tree so far:


Now complete the tree, drawing the number of options for the remaining position in each case:


The number of ways to choose the players is equal to the number of boxed outcomes on the right.
Total ways = 5.

The correct answer is B.

[Scott@TargetTestPrep]

Seven students are trying out for the school soccer team, on which there are three available positions: fullback, sweeper, and goalie. Each student can only try out for one position. The first two students are trying out for fullback. The next two students are trying out for sweeper. The remaining three students are trying out for goalie. However, the fourth student will only play if the second student is also on the team, and the third student will only play if the fifth student is on the team. How many possible combinations of students are there to fill the available positions?

A 3
B 5
C 7
D 10
E 12
OA is b

what is the best mathematical approach to use here? can any expert help?

Thanks for responding to my need

If the fullback is the first student, then the sweeper must be the third student (since the fourth student won't play unless the fullback is the second student). Since the sweeper is the third student, goalie must be the fifth student (since the third student won't play unless goalie is the fifth student). Thus, we see that there is only one way to fill the positions if the fullback is the first student.

If the fullback is the second student, the sweeper can be the third or fourth student.

If the sweeper is the third student, the goalie must be the fifth student (same reason as above); therefore we see that there is only one way to fill the positions if the fullback is the second student and the sweeper is the third student.

If the sweeper is the fourth student (which is possible since the fullback is the second student), then the goalie can be any one of the remaining three students. Therefore, we see that there are three ways to fill the positions if the fullback is the second student and the sweeper is the fourth student.

In total, there are 1 + 1 + 3 = 5 ways to fill the positions.

Answer: B