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Probability Level pending

How many 15 letter arrangements of 5 A's , 5 B's , 5 C's have no A's in the first 5 letters,no B's in the next 5 letters , and no C's in the last 5 letters ?


The answer is 2252.

Nipun Gupta by Nipun Gupta , 23, India

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1 solution

If first block contains k k B's and 5 − k 5-k C's, remaining 5 − k 5-k B's must be in third block and remaining k k C's must be in second block.

So third block contains k k A's and 5 − k 5-k B's, second block contains k k C's and 5 − k 5-k A's.

For each k k , the numbers of the valid arrangements is ( 5 k ) 3 \displaystyle\binom{5}{k}^3 .

So, the total number of the valid arrangements is ∑ k = 0 5 ( 5 k ) 3 = 2252 \displaystyle\sum_{k=0}^{5}\binom{5}{k}^3=\boxed{2252} .

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