2008 AMC 10B Problems/Problem 22

Problem

Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?

A) 1/12 B) 1/10 C) 1/6 D) 1/3 E) 1/2

Solution

There are two ways to arrange the red beads.

1) R _R _ R _ 2) R _ _ R _ R

In the first, there are three ways to place a bead in the first space, two for the second space, and one for the third, so there are 6 arrangements. In the second, a white bead must be placed in the third space, so there are two possibilities for the third space, two for the second, and one for the first. That makes 4 arrangements. There are 6+4=10 arrangements in total. The two cases above can be reversed, so we double 10 to 20 arrangements. Also, in each case, there are three ways to place the first red bead, two for the second, and one for the third, so we multiply by 6 to get 20*6 = 120 arrangements. There are 6! = 720 total arrangements so the answer is 120/720 = 1/6 (C)