Balls, balls and balls
Probability
Level
3
There are 2 identical white balls , 3 identical red balls and 4 green balls of different shades. The number of ways in which they can be arrranged in a row such that atleast one ball is separated from the balls of the same colour is :
The answer is 30096.
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1 solution
It is obviously easier to calculate the complement in this case: The number of arrangements where at least one ball is separated from the balls of the same color = The difference between the total number of arrangements of all the 9 balls and the number of arrangements where the balls of the same color are put together.
I. The total number of arrangements of the 9 balls --- Since there are 2 identical white balls and 3 identical red balls, this is a case of permutations with repetition . The answer is: 2 ! ∗ 3 ! 9 ! = 30240
II. The total number of arrangements where balls of the same color are together --- Since the white and red balls are identical, there is just one way to arrange either color. There are, however, 4! arrangements for the balls with the green color. Once the balls of the same color are put together, there are 3! ways of arranging them because there are 3 different colors. The answer is: 3!*4! = 144
Therefore, the number of arrangements where at least one ball is separated from balls of the same color is: 30240 - 144 = 3 0 0 9 6