Answer to Question #142104 in Discrete Mathematics for Promise Omiponle

Question #142104
Let S be a finite set, with |S|= 200. Find the number of subsets of S containing more than 2 elements
1
Expert's answer
2020-11-16T19:52:32-0500

The number of subsets of SS containing k elements is equal to (nk)=n!k!(nk)!\left(\begin{array}{c}n\\k\end{array}\right)=\frac{n!}{k!(n-k)!}, where n=S=200.n=|S|=200.


Firstly, find the number mm of subsets of SS containing not more than 2 elements. The emptyset is a unique set containg no elements. There are 200 different subsets of SS having one element. The number of subsets containg two elements is (2002)=200!2!198!=200199198!2198!=100199=19900.\left(\begin{array}{c}200\\2\end{array}\right)=\frac{200!}{2!\cdot198!}=\frac{200\cdot199\cdot198!}{2\cdot 198!}=100\cdot199=19900.

Then m=1+200+19900=20101.m=1+200+19900=20101. Since the set SS contains 22002^{200} subsets, the number of subsets of SS containing more than 2 elements is 2200m=220020101.2^{200}-m=2^{200}-20101.



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