Question

Number of necklaces can be made using 5 identical
red beads and 2 identical blue beads is
(1) 720
(2) 360
(3) 3
(4) 2​

Answer 1

Step-by-step explanation:

There are (82) ways to arrange the 6 red beads and 2 blue beads along a circle. The dihedral group D16 of order 16 acts on the set of arrangements. Let ω be a rotation about the center by angle 2π8, σ, reflection about the line joining the mid points of a pair of opposite sides (there are 4 such reflections), τ a reflection about the line joining a pair of opposite vertices (there are four such reflections).

For ω,ω2,ω3,ω5,ω6,ω7 there are no fixed elements. For ω4 four elements are fixed. For any σ, 4 elements are fixed and for any τ, 4 elements are fixed. Thus by Burnside Theorem, the number of distinct necklaces is

116(28+4+16+16)=4