Problem 7 · Inclusion–exclusion with rotational symmetry · 5 points
Made by AI and not reviewed by a real person.
How many circular arrangements of the letters AABBCCDD have no identical letters next to each other? Count rotations as the same, but do not identify mirror images.
- 96
- 744
- 315
- 93
- 318
Show a hint
- First count arrangements on eight numbered seats. Then ask whether every valid arrangement produces eight different rotations.
- To count on numbered seats, use inclusion–exclusion: temporarily glue together any equal-letter pairs that are required to be adjacent.
Show the answer and solution
Answer: A — 96
Choose k of the four equal-letter pairs to be adjacent. Gluing those pairs into blocks gives 8(7−k)!/2^(4−k) arrangements on numbered seats. Inclusion–exclusion therefore gives 2520−2880+1440−384+48=744 valid arrangements. Of these, 4!=24 repeat after a half-turn: their first four letters contain A, B, C and D once each. Every other valid arrangement has eight distinct rotations, while these 24 have only four. The answer is (744−24)/8+24/4=96. The tempting answer 93 comes from dividing all 744 by eight, overlooking the half-turn arrangements.