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Consider the set |n| = {1,2,...,n}. How many subsets does it have of cardinality k and that contain the element 1?

I understand that with each element, you can either include it or not to have a total of k elements, but I'm not sure exactly how to show this for any set of finite size.

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    Perhaps the most basic way to show this is by induction: the number of subsets with $\,k\,$ elements that a set with $\,n\, $elements, $\,n\geq k\, $ , is given by the binomial coefficient $$\binom{n}{k}:=\frac{n!}{k!(n-k)!}$$2012-12-09

2 Answers 2