Question

Goofus and Gallant try out for the quiz team. Goofus, Gallant, and seven of their classmates are trying out for the quiz bowl team. There are four positions available on the team.

(a) If each position is different, how many ways are there that the team can be made?
(b) If each position is the same, how many ways are there that the team can be made?
(c) Suppose each position is the same. How many teams include both Goofus and Gallant?
(d) Suppose each position is the same. How many teams include Goofus or Gallant? ("Or" means "at least one of.")

Answer 1

Along with Goofus and Gallant, there are nine classmates for quiz bowl team and there are four positions available on the team.

(a) A team of 4 members can be selected from 9 members in ways and each position is different, the selected 4 members can be arranged in 4! ways.

The number of ways a team of 4 can be made if each position is different are = 3024 ways.

(b) A team of 4 members can be selected from 9 members if each position is same in ways = 126 ways.

(c) In this case both Goofus and Gallant must include in team. So the remaining 2 members (in 4 members team) from the remaining 7 classmates (from 9 classmates except Goofus and Gallant) can be selected in ways.

Suppose each position is the same, the number of teams include both Goofus and Gallant are = 21 teams.

(d) In this case teams include Goofus or Gallant i.e., atleast one of them must present in the team. So, one member is fixed in the team and the remaining 3 members (in 4 members team) from the remaining 8 classmates (from 9 classmates except Goofus or Gallant) can be selected in ways.

Suppose each position is the same, the number of teams include Goofus or Gallant are = 56 teams.