Problem Solving

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

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.