Beads in necklace

Probability Level 3

The number of ways in which 5 beads of different colors can form a necklace is

12 20 120 24

Arpit Dhimanj by Arpit Dhimanj , 23, India

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1 solution

In general the n n beads of different colours to make a necklace can be arranged in ( n − 1 ) ! (n -1)! ways. As the left and right arrangements are same for the necklace then the number of arrangements will be ( n − 1 ) ! / 2 (n -1)!/2 we are given n = 5 n=5 so the arrangement must be ( 5 − 1 ) ! / 2 = 4 ! / 2 = 12 (5 -1)!/2 = 4!/2 = 12

3 Helpful 0 Interesting 0 Brilliant 0 Confused

The rotational symmetry overcounts the number by 5 5 times so we divide 5 ! 5! by 5 5 .

The reflectional symmetry overcounts the number by 2 2 times so we divide the result by 2 2 to get 12 \boxed{12} as the final answer.

Daniel Liu - 6 years, 11 months ago

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@Daniel Liu I didn't realize that we could use the rotational symmetry and reflectional symmetry in this problem.Thanks it is always good to learn another method.

Mardokay Mosazghi - 6 years, 11 months ago

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