Selecting the Divisional Quiz Team
Seven students are shortlisted for a four-member team to represent the school at the divisional quiz: Ayesha (Class 10, science club), Babul (Class 9, debate club), Chandni (Class 10, science club), Dipu (Class 9, science club), Eshita (Class 10, debate club), Farhan (Class 9, debate club) and Gita (Class 10, debate club). The selection rules are: (i) exactly four members; (ii) at least one member from Class 9 and at least one from Class 10; (iii) no more than two science-club members; (iv) Ayesha and Dipu cannot both be selected; (v) if Farhan is selected, Chandni must be selected. The teacher-in-charge proposes four teams and asks the club to check them instead of voting: Team P - Ayesha, Babul, Eshita, Farhan. Team Q - Ayesha, Chandni, Dipu, Babul. Team R - Ayesha, Chandni, Eshita, Farhan. Team S - Babul, Dipu, Eshita, Gita. No substitution is allowed after the list is submitted, and the principal wants the reasoning written down before she signs.
1. Test each of the four proposed teams against all five rules and state which are valid.
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Team P (Ayesha, Babul, Eshita, Farhan): Class 9 = Babul, Farhan ✓; Class 10 = Ayesha, Eshita ✓; science = Ayesha only, 1 ≤ 2 ✓; rule (iv) ✓; rule (v) fails, because Farhan is selected and Chandni is not. Invalid.
Team Q (Ayesha, Chandni, Dipu, Babul): Class 9 = Babul, Dipu ✓; Class 10 = Ayesha, Chandni ✓; science = Ayesha, Chandni, Dipu = 3, which breaks rule (iii); and Ayesha and Dipu are both present, breaking rule (iv). Invalid on two counts, so fixing only one would not rescue it.
Team R (Ayesha, Chandni, Eshita, Farhan): Class 9 = Farhan ✓; Class 10 = Ayesha, Chandni, Eshita ✓; science = Ayesha, Chandni = 2 ✓; rule (iv) ✓ since Dipu is absent; rule (v) ✓ since Farhan is accompanied by Chandni. Valid.
Team S (Babul, Dipu, Eshita, Gita): Class 9 = Babul, Dipu ✓; Class 10 = Eshita, Gita ✓; science = Dipu only ✓; rule (iv) ✓ since Ayesha is absent; rule (v) ✓ since Farhan is absent. Valid.
So two teams pass, and the choice between them is a judgement about balance, not a rule breach.
2. A student claims that a valid team might contain no debate-club member at all. Prove or disprove this.
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The claim is false, and it can be disproved without listing teams.
Count the non-debate students. The debate club contains Babul, Eshita, Farhan and Gita - four people. The remaining three, Ayesha, Chandni and Dipu, are all science-club members, so the pool of students who are not in the debate club has exactly three members.
Rule (i) requires four members. A team with no debate-club member would have to be drawn entirely from a pool of three people, which is impossible. Therefore every valid team, indeed every conceivable four-person team from this shortlist, contains at least one debate-club member.
Note how strong the result is: it uses only the roster sizes, not rules (ii)–(v). That means the conclusion survives any change to the class-mix rule or the Ayesha–Dipu rule. A good check is to try to break it - pick the three non-debate students and look for a fourth; whatever fourth name is added is by definition a debate-club member.
3. Can a valid team contain both Farhan and Dipu? If so, give every such team and say whether the answer is unique.
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Yes. Rule (v) forces Chandni in alongside Farhan, so the team is {Farhan, Dipu, Chandni} plus one more member.
Check the fixed three: Class 9 = Farhan, Dipu ✓; Class 10 = Chandni ✓; science = Dipu and Chandni = 2, already at the rule (iii) limit; rule (iv) is safe because Ayesha is absent.
Because the science quota is full, the fourth member must come from the debate club: Babul, Eshita or Gita. Ayesha and any other science student are excluded. Each choice keeps Class 9 (Farhan, Dipu) and Class 10 (Chandni plus the fourth) represented and breaks no rule.
That gives exactly three valid teams: {Farhan, Dipu, Chandni, Babul}, {Farhan, Dipu, Chandni, Eshita} and {Farhan, Dipu, Chandni, Gita}. The answer is not unique, and no additional rule in the list can choose between them - the tie-break must come from outside the rulebook, such as quiz-subject strengths or prior performance, and should be stated openly rather than presented as forced by the rules.