Problem 6 · Count cyclic arrangements using gaps and rotational symmetry · 4 points
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A necklace has 4 black beads and 8 white beads, with no two black beads touching. How many arrangements are possible if two arrangements count as the same exactly when one is a rotation of the other?
- 9
- 10
- 8
- 35
- 105
Show a hint
- Look at the numbers of white beads between successive black beads. What happens to that list when you choose a different black bead as your starting point?
- Count the positive four-part sums, then check which lists remain unchanged when shifted by one, two, or three places.
Show the answer and solution
Answer: B — 10
The four gaps between successive black beads contain positive numbers of white beads summing to 8, giving C(7,3) = 35 ordered lists. Under cyclic shifts of a list, 35 lists are unchanged by a shift of zero places; 1 is unchanged by a shift of one place; 3, of the form (a,b,a,b) with a+b=4, are unchanged by a shift of two places; and 1 is unchanged by a shift of three places. Each distinct necklace contributes four across these fixed-list counts, so the answer is (35+1+3+1)/4 = 10. The tempting answer 35 counts the same necklace again when a different black bead is chosen as the starting point.