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Out of 15 balls, of which some are white...

Out of 15 balls, of which some are white and the rest are black, how many should be white so that the number of ways in which the balls can be arranged in a row may be the greatest possible? It is assumed that the balls of same color are alike?

Text Solution

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The correct Answer is:
7 or 8
Let there be r white and (15-r) black balls.
Then, total number of permutations of these balls is `15!//r!(15-r)!= .^(15)C_(r )` since r white balls are alike and (15-r) black balls are alike.
Therefore, the number of arrangement is `.(15)C_(r )` which is maximum, when `r=(15-1)//2 " or " (15+1)//2`, i.e., r=7 or 8.
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