Question

Mary has a string of n empty beads to make jewelry from, where n=2k is some even positive integer. She has an extremely large supply of purple, blue, orange, and red light bulbs. She wants to place the beads in the bracelet in a symmetric way, meaning the order of colors should be the same left-to-right and right-to-left. How many different symmetric ways are there for Mary to arrange n beads of those four colors on this string?

Answer 1

n/2 beads needs to be arranged with any of the 4 colors. The other n/2 beads are to be arranged with corresponding colors.

(i.e The bead no. 1 and bead no. n of the same color, bead no. 2 and bead no. n-1 of the same color)

The required number of symmetric ways =