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How many ways are there to place 7 differently colored toys into 5 identical urns if the urns can be empty? Note that all balls have to be used.(a) 320(b) 438(c) 1287(d) 855I have been asked this question in quiz.My question is from Counting in division Counting of Discrete Mathematics

Answer»

The correct answer is (d) 855

For explanation: The problem can be DESCRIBED as distinct objects into any number of identical bins and this number can be found with B7 = ∑S(7,k), where S(7,k) is the number of distributions of 5 distinct objects into k identical non-empty bins, so that S(7,1) = 1, S(7,2) = 63, S(7,3) = 301, S(7,4) = 350 and S(7,5) = 140. These values can be found using the recurrence relation identity for Stirling numbers of the second KIND. Thus, B7 = 1 + 63 + 301 + 350 + 140 = 855.



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