Question

Lucy has a necklace which has red beads and blue beads. For every red bead, there are four blue beads. There are 20 beads in total. How many of these are blue?

Solve with Steps Ratio and Proportion​

Answer 1

Step-by-step explanation:

We use the Polya Enumeration Theorem (PET). The cycle index Z(D6) of the dihedral group D6 is given by

Z(D6)=112⎛⎝∑d|6φ(d)a6/dd+3a32+3a21a22⎞⎠.

This is

112(a61+a32+2a23+2a6+3a32+3a21a22)=112(a61+2a23+2a6+4a32+3a21a22).

We seek

[R2B2YG]Z(D6;R+B+Y+G).

The only contribution comes from

112(a61+3a21a22).

This is because with the Polya substitution the terms 2a23+2a6+4a32 produce powers of three, six, and two, exclusively.

Continuing, we get

112[R2B2YG](R+B+Y+G)6+14[R2B2YG](R+B+Y+G)2(R2+B2+Y2+G2)2.

This is

112(62,2,1,1)+14×2×2=16.

BTW the convention at the OEIS is to use the term necklace for cyclic symmetries and bracelet for dihedral ones.