Green Rainbow

Posted on Tuesday, July 14th, 2009 at 9:16 pm

Seven poker chips, one of each color of the rainbow (red, orange, yellow, green, blue, indigo, and violet), ar?

need help!
Seven poker chips, one of each color of the rainbow (red, orange, yellow, green, blue, indigo, and violet), are stacked randomly in a pile. What is the probability that the blue and green chips are not adjacent?

71.43%

The total number of combinations that the chips can be stacked in is 7! or in other words 7x6x5x4x3x2x1 = 5040 different ways. This is because.
The bottom chip can be any of the 7
The bottom but one can be any of the 6 remaining (one is already in place)
The bottom but 2 can be any of the 5 remaining (the bottom two are already placed)
The bottom but 3 can be any of the 4 remaining (the bottom three are already placed)
and so on
Therefore the chips can be arranged in 7x6x5x4x3x2x1 = 7! = 5040 ways.

Now, if the blue and green chips are adjacent lets treat them as a single chip, so there are 6 "chips" not 7 ie Red, Orange, Yellow, GREENBLUE, Indigo and Violet.
These chips can be arranged in 6! ways. In a similar method to above, the
Ist = one of 6
2nd = one of 5 etc etc
Therefore they can be arranged in 6x5x4x3x2x1 = 720 different ways.
Then we have to think, all these ways have "green" on the same side compared to "blue" eg green is always on top of blue. However blue can also be on top of green to be adjacent. Therefore there are another 6! or 720 combinations of BLUEGREEN rather than "greenblue"
Therefore there are 720 + 720 = 1440 combinations where blue and green are next to each other.
Therefore of the total 5040 combinations 1440 have green and blue adjacent and 5040 - 1440 = 3600 combinations where the two colours are not adjacent.

Therefore the probability is 3600 / 5040 = 71.43% to two decimal places