Assignment:
Q1. 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 (1 - x - x2 - x3 - x4 - x5 - x6)-1 is the generating function for the number of ways a sum of r can occur if a die is rolled any number of times.
Q2. A) Find the exponential generating function for Sn,r the number of ways to distribute r distinct objects into n distinct boxes with no empty box. Consider n a fixed constant.
B) Determine Sn,r / n! is called a Stirling number of the second kind.
Provide complete and step by step solution for the question and show calculations and use formulas.