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# Combinations and Permutations : Ways to paint the 10 identical rooms 5 colors; Find the egf for the number of ways to distribute r distinct objects into n boxes.

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Exercise #1

A) How many ways are there to paint the 10 identical rooms in a hotel with five colors if at most three rooms can be painted green, at most three painted blue, at most three red, and no constraint on the other two colors, black and white.
B) Show that ...is the generating function for the number of ways a sum of r can occur if a die is rolled any number of times.

Exercise#2

A) Find the exponential generating function for the number of ways to distribute r distinct objects into n distinct boxes with no empty box. Consider n a fixed constant.
B) Determine is called a Stirling number of the second kind.

Please see the attached file for the fully formatted problems.

https://brainmass.com/math/combinatorics/51494

#### Solution Summary

Ways to paint the 10 identical rooms 5 colors are investigated. The egf for the number of ways to distribute r distinct objects into n boxes is investigated. The solution is detailed and well presented.

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