Question

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) 855

I have been asked this question in quiz.

My question is from Counting in division Counting of Discrete Mathematics

Answer 1

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.