Question

12) How many ways are there to distribute 150 identical balls to 12 distinct boxes, 6 red boxes labelled 1 to 6 and 6 blue boxes labelled 1 to 6, such that the blue be with label i and the red box with labeli receive the same number of balls, for every ie (1,2,3,4,5, 6}? [10 marks

Answer 1

Since the blue and the red boxes have to get the same number of balls for all , it is enough to count the number of ways there are to distribute
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balls into 6 red boxes. Once this is done the same configuration can be duplicated on the blue boxes. Since the balls are all identical, there is no need to count the permutation of the balls.

Now look at the number of ways we can distribute 75 balls into 6 different red boxes. To do this, think of the balls as arranged in a straight line. To partition, we need to pick 5 gaps between the balls. Since blanks are allowed we need to add extra blanks (4 of them) before and after the line. There are blanks and we have to choose 5 of them.

745584

Hence the number of ways to partition is .

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