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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Tags: combinations, combinatorics, sets, Stirling Numbers, subsets

This entry was posted on November 12, 2012 at 4:37 pm and is filed under General interest, Honors Alg. 2, Precalculus, Statistics Readings. You can follow any responses to this entry through the RSS 2.0 feed.
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