The Rule That Does No Work
Seven volunteers - A, B, C, D, E, F and G - are available for a four-member survey team to be sent to a hill district. The district office's rules are fixed: (i) if A is selected, F must be selected; (ii) B and C cannot both be selected; (iii) exactly one of D and E must be selected; (iv) if G is selected, C must be selected; (v) A and G cannot both be selected. The team must have exactly four members. The block officer has circulated rule (v) to the panchayats as the "hill-terrain safeguard". A panchayat secretary objects that rule (v) does no work at all and should be struck out.
1. How many valid four-member teams exist? List them and show the enumeration that produces them.
Read the worked answer
Rule (iii) fixes one seat: every team contains exactly one of D, E. The remaining three places must therefore come from {A, B, C, F, G}. Enumerate the ten possible triples and test each against (i), (ii) and (iv):
| Triple | Test | Verdict |
|---|---|---|
| A B F | A→F ✓; B,C not together ✓ | valid |
| A C F | A→F ✓; B,C not together ✓ | valid |
| A B G | A→F ✗; G→C ✗ | invalid |
| A C G | A→F ✗ | invalid |
| A F G | G→C ✗ | invalid |
| B C F | B and C together ✗ | invalid |
| B C G | B and C together ✗; G→C ok | invalid |
| B F G | G→C ✗ | invalid |
| C F G | G→C ✓; B,C not together ✓ | valid |
| B C A | (same as A B C) B,C together ✗ | invalid |
Three surviving triples - ABF, ACF, CFG - each paired with D or E gives six teams:
- A B F D 2. A B F E 3. A C F D 4. A C F E 5. C F G D 6. C F G E
Note the trap: it is tempting to list triples without checking whether D/E consume a place. Anyone who forgets rule (iii) will overcount by roughly a factor of two.
2. Evaluate the panchayat secretary's claim that rule (v) does no work. Give a general argument, not just a check of your list.
Read the worked answer
The claim is correct; rule (v) is redundant.
General proof: suppose a team contained both A and G. Then (i) forces F in, and (iv) forces C in. That is already four named members - A, G, F, C - with no room for the mandatory single member from {D, E}, because rule (iii) requires exactly one of them and the team is capped at four. So no admissible team can contain both A and G; rule (v) forbids a combination that (i), (iii) and (iv) already forbid.
The enumeration confirms this: G appears only in the CFG triple, which contains C and not A.
Two consequences for practice. First, a rule can be perfectly true and still be inert; its presence creates a false impression of an extra safeguard and invites the officer to defend it as if it were load-bearing. Second, redundancy is design information, not a defect - the correct response is to record why it is redundant (the four-seat cap interacting with the A→F and G→C implications) rather than to delete it silently. If the team size were ever raised to five, rule (v) would immediately become binding: A G F C plus one of D/E is exactly a five-member team that (v) alone would exclude. So the rule should be re-examined, not struck out, whenever the size changes.
3. The officer now wants to maximise hill-terrain experience. Only A, C and E have it. Which valid team should be sent, and is the choice unique?
Read the worked answer
Score each of the six valid teams by counting members among {A, C, E}:
- A B F D → A only = 1
- A B F E → A, E = 2
- A C F D → A, C = 2
- A C F E → A, C, E = 3
- C F G D → C = 1
- C F G E → C, E = 2
Send A, C, F and E. It is the only team reaching 3, so the choice is unique among the six admissible teams.
Two cautions before this is written up. First, the constraint set - not the scoring - is doing the work: the maximum is 3 rather than 4 only because including E excludes D by rule (iii), and D is the only non-experienced member common to all three-member-experience options. Second, if the officer wants all three experienced people and G, the rules make it impossible at four members, so the honest recommendation is either to accept 3 and field the A C F E team, or to seek a relaxation of rule (iii) rather than to quietly drop rule (v) - which, as shown, would change nothing.