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Seven students are trying out for the school soccer team, on

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Seven students are trying out for the school soccer team, on

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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

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alanforde800Maximus wrote:
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.

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alanforde800Maximus wrote:
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
$$\left. \matrix{
{\rm{Full}}\,\,\,{\rm{:}}\,\,\,\,{\rm{2}}\,\,{\rm{students}}\,\,\left( {A,B} \right) \hfill \cr
{\rm{Swee}}\,\,\,{\rm{:}}\,\,\,{\rm{2}}\,\,{\rm{students}}\,\,\left( {C,D} \right) \hfill \cr
{\rm{Goal}}\,\,\,{\rm{:}}\,\,\,{\rm{3}}\,\,{\rm{students}}\,\,\left( {E,F,G} \right)\,\, \hfill \cr} \right\}\,\,\,\,\,{\rm{with}}\,\,{\rm{restrictions}}\,\,\,\left\{ \matrix{
\,D\,\,\, \Rightarrow \,\,\,B\,\,\,\left( * \right) \hfill \cr
\,C\,\,\, \Rightarrow \,\,\,E\,\,\,\left( {**} \right) \hfill \cr} \right.$$
$$?\,\,\,:\,\,\,\# \,\,\left( {{\rm{Full}}\,,\,\,{\rm{Swee}}\,,\,{\rm{Goal}}} \right)\,\,{\rm{choices}}$$

$${\rm{?}}\,\,\,{\rm{:}}\,\,\,{\rm{manual}}\,\,\underline {{\rm{organized}}} \,\,{\rm{work}}\,\,{\rm{technique}}\,\,\,\,\left\{ \matrix{
\,\left( {A,C,E} \right)\,\,\,\left( {**} \right)\,\,\,1. \hfill \cr
\,\left( {A,D,{\rm{no!}}} \right)\,\,\,\left( * \right) \hfill \cr
\,\left( {B,C,E} \right)\,\,\,\left( {**} \right)\,\,\,2. \hfill \cr
\,\left( {B,D,E} \right)\,\,\,\,\left( * \right)\,\,\,\,3.\, \hfill \cr
\,\left( {B,D,F} \right)\,\,\,\,\left( * \right)\,\,\,\,4. \hfill \cr
\,\left( {B,D,G} \right)\,\,\,\,\left( * \right)\,\,\,\,5. \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,? = 5$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Hi alanforde800Maximus,

This question uses what's called "Formal Logic" (a concept that you would see repeatedly on the LSAT, but rarely on the GMAT). You can answer it with some drawings and careful note-taking.

Based on the information in the prompt, we have seven players (A,B,C,D,E,F,G). We're asked for the total groups of 3 that can be formed with the following restrictions:

1) Only 1 player per position
2) A and B are trying out for fullback
C and D are trying out for sweeper
E, F and G are trying out for goalie
3) D will only play if B is also on the team
C will only play if E is also on the team

The big restrictions are in the two 'formal logic' rules....
-We can put E with ANYONE, but we can only put in C if E is ALSO there.
-We can put B with ANYONE, but we can only put in D if B is ALSO there.

By extension....
D can NEVER be with A (because then B would not be in the group)
C can NEVER be with F or G (because then E would not be in the group)

As such, there are only a few possibilities. We can have...
ACE
BCE
BDE
BDF
BDG

Final Answer: B

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