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Distribution into Bins

How can five candies be distributed among 3 friends? Well, the answer depends on if you can tell the candies apart!

Distinct Objects in Identical Bins

         

Bernhard is organizing his comic book collection. He has 12 distinct comic books that he would like organize into 2 non-empty groups.

How many ways are there for Bernhard to organize his comic book collection?

Pierre has a collection of 10 distinct action figures. He would like to show them off in 3 identical displays. He would like each display to have at least 2 action figures, but no more than 4 action figures.

How many ways are there for Pierre to organize his action figure displays?

How many ways are there to distribute 8 distinct clay pots into 5 identical shelves such that no shelf is left empty?

For your reference, the table below shows the values of Stirling numbers of the second kind for \(n,r\le 7\).

\[ \begin{array} { c | c c c c c c } _n\backslash ^r & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 0 & 1 \\ 1 & 0&1\\ 2& 0&1&1\\ 3& 0&1&3&1\\ 4& 0&1&7&6&1\\ 5& 0 &1&15&25&10&1 \\ 6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 \\ 7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 \\ \end{array} \]

There are 2 red identical boxes and 2 blue identical boxes.

How many ways are there to distribute 6 distinct objects into these boxes such that no box is left empty?

How many ways are there to distribute 10 distinct students into 4 groups such that each group has at least 2 students in it?

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