The four-member audit team and five written rules
Eight officers - Anita, Bhavna, Chetan, Divya, Esha, Farhan, Gopal and Hari - are being considered for a four-member audit team. Anita, Bhavna, Divya and Esha are women; the other four are men. The branch head's written rules are: (i) exactly one of Anita and Chetan must be on the team; (ii) Divya and Farhan cannot serve together; (iii) if Esha is on the team, Bhavna must also be on it; (iv) at least one of Gopal and Hari must serve; (v) the team must include at least two women. A team is 'valid' only if it satisfies all five rules, has exactly four members, and repeats no member. The branch head has proposed Anita, Divya, Farhan and Hari and wants to know, before signing, whether the proposal survives and what the alternatives are.
1. Test the proposed team of Anita, Divya, Farhan and Hari against each rule and state whether it can be signed.
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Check rule by rule:
- (i) Exactly one of Anita and Chetan: Anita is in, Chetan is out → satisfied.
- (ii) Divya and Farhan not together: both are in → violated.
- (iii) Esha implies Bhavna: Esha is absent, so the conditional is not triggered → satisfied.
- (iv) At least one of Gopal and Hari: Hari is in → satisfied.
- (v) At least two women: Anita and Divya → satisfied.
Cannot be signed. Only rule (ii) fails. The cheapest repair is a single swap: replace Farhan with Chetan, Gopal or Bhavna. Replacing Farhan with Chetan would break rule (i) (it would put both Anita and Chetan in), so the workable one-member swaps are Farhan → Gopal or Farhan → Bhavna, giving Anita, Divya, Gopal, Hari or Anita, Divya, Bhavna, Hari; both keep two or more women and satisfy (i), (ii) and (iv).
2. How many valid teams contain both Esha and Gopal? Show the enumeration rather than guessing.
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Rule (iii) forces Bhavna in alongside Esha. The team so far is Esha, Gopal, Bhavna, plus one more member X.
- Rule (iv): Gopal already satisfies it, so it places no restriction on X.
- Rule (i): exactly one of Anita and Chetan must be in, and neither is yet in, so X must be Anita or Chetan.
Test the possible X:
| X | Result |
|---|---|
| Anita | Women = Anita, Bhavna, Esha (3); no Divya–Farhan clash → valid |
| Chetan | Women = Bhavna, Esha (2); rule (i) holds as Anita is out → valid |
| Divya | Neither Anita nor Chetan in → fails (i) |
| Farhan | Fails (i) |
| Hari | Fails (i) |
Valid teams: {Anita, Bhavna, Esha, Gopal} and {Bhavna, Chetan, Esha, Gopal}. Answer: 2.
3. If rule (iv) were deleted, would your count in the previous task change? Explain the reasoning, not just the number.
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No - the count stays at 2.
Rule (iv) is already satisfied in every candidate considered, because Gopal is a fixed member of every team being counted. Removing a rule that no candidate ever violates cannot remove a candidate.
The general check is worth stating: a rule only changes a count if at least one otherwise-eligible team fails it. Here rule (iv) is inert; rule (iii) was also inert in the same way, except that it forced Bhavna in, which did restrict the field. The rules that actually shaped the answer were (i), which fixed the fourth seat, and (iii), which fixed the third.