Probability
# Distribution into 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?