What are Stirling Numbers of the Second Kind?

How are the following two questions different?

1.)  How many ways can I select k elements from a set of n elements?

2.)  How many ways can I select k subsets from a set of n elements?

The two questions are totally different but a speedy reading could miss the point.  In the first question, we are counting subsets of k elements each taken from n elements, and in the second question, we a counting how many different ways we can choose k sets (which can have different numbers of elements,) from n elements.  The first answer is arrived at using combination theory; n choose k.

The second question involves Stirling Numbers of the Second Kind which can be calculated, or alternatively looked up in published tables for various n‘s and k‘s.

Thank you to Martin Griffiths for his article in Mathematics Teacher magazine (Nov. 2012, pg.317) for bringing this concept to the fore.  See his article for a more in-depth discussion: “Close Encounters with Stirling Numbers of the Second Kind.”

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