Problem:
Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
Answer Choices:
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Solution:
There are distinguishable orders of the beads on the line. To meet the required condition, the red beads must be placed in one of four configurations: positions , and , positions , and , positions , and , or positions , and . In the first two cases, the blue bead can be placed in any of the three remaining positions. In the last two cases, the blue bead can be placed in either of the two adjacent remaining positions. In each case, the placement of the white beads is then determined. Hence there are orders that meet the required condition, and the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions