A small probability class of fifteen students gets to work o

A small probability class of fifteen students gets to work on some problems. If there are five different problems, and each student has the problems, how many possible outcomes are there? Now the professor decides to divide the students up into five (unordered) working groups of 3 (unordered) students each. Now how many outcomes are there? The professor puts a bowl of 40 identical jellybeans at the front of the class. If each student gets at least one jellybean, how many different distributions are possible?

Solution

a)

There are 5^15 = 30517578125 possible outcomes. [ANSWER]

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b)

There are, by permutation of like objects,

#ways = 15!/(3!3!3!3!3!) = 168168000 [ANSWER]

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c)

We first give one jellybean to each person.

Then we have n = 25 beans to be distrbuted to r = 15 people.

By the balls and urns problem, the number of ways to distribute 25 beans to 15 people is

#ways = (n+r-1)!/[(r-1)!n!] = 15084504396 [ANSWER]