| A Row of Marbles |
| A boy has six yellow marbles and four blue marbles, each distinct from the others. He arranges his ten marbles in a row. What is the probability that no blue marble lies next to another? |
Problem Moderated by: Douglas |
| Problem Solution |
Consider the six yellow marbles, as they will be arranged, as follows:
Y1-Y2-Y3-Y4-Y5-Y6
Next to each yellow marble, there will be either zero blue marbles or one blue marble, and a total of four. The arrangement, then, looks like this, where there could be a blue marble where each B is.
B-Y1-B-Y2-B-Y3-B-Y4-B-Y5-B-Y6-B
There are seven different possible places for a blue marble to lie; as such, there are 7*6*5*4 ways for four of these slots to be chosen. Finally, there are 6! ways for the six yellow marbles to be ordered, and 10! ways for all of the ten marbles to be ordered. The probability that no blue marble lies next to another is therefore
(6!*7*6*5*4)/10! = 7*6*5*4/10*9*8*7=1/6 |
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