Alternating in a Row

Question

Three traders and three interns line up in a row of 6 for a desk photo. House rule: no two traders may stand next to each other, and no two interns may stand next to each other. How many lineups are possible?

Accepted Answer

Pattern first, people second. Ignore names and look only at the role sequence. With 3 T's and 3 I's in a row, "no two alike adjacent" forces strict alternation: once two consecutive slots share a role, the condition fails, and any non-alternating sequence of 3 and 3 must have such a pair (six slots with no two alike adjacent means roles flip at every step). So the only role patterns are TITITI quad textand quad ITITIT. Now fill each pattern with actual people: the three traders arrange in their three slots in 3! ways and the interns in 3! ways: 2 cdot 3! cdot 3! = 2 cdot 6 cdot 6 = mathbf72. The trap: jumping straight to arranging people and trying to subtract bad lineups from 6! = 720 by inclusion-exclusion. It works but takes ten times longer; recognizing that the skeleton of the arrangement is forced collapses the problem to a multiplication. Always ask "what does the constraint do to the shape?" before touching the labels. The factor-the-problem move (pattern count times fillings per pattern) is the same decomposition behind seating couples, arranging letters with forbidden adjacencies, and any problem where roles and identities can be chosen in separate stages.

Alternating in a Row

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