How many subsets with an odd number of elements does a set with 10 elements have

skeexerxo175o

skeexerxo175o

Answered question

2021-11-14

How many subsets with an odd number of elements does a set with 10 elements have?

Answer & Explanation

Egreane61

Egreane61

Beginner2021-11-15Added 16 answers

Definitions
An r-combination of a set of n elements is a subset that contains r of the n elements. The number of r-permutations of a set with n elements is C(n,r)=n!r!(nr)! with nn(n1)21
Solution:
We are interested in the number od subset with an odd number of elements from a set with 10 elements.
n=10
r=1,3,5,7,or9
A subset of r elements from a set with n elements is an r-combinations and thus we are intersted in the number of r-combinations of a set with n=10 elements that contain an odd number of elemets.
Number of subsets
=r{1,3,5,7,9}C(10,r)
=c(10,1)+C(10,3)+C(10,5)+C(10,7)+C(10,9)
=10!1!(101)!+10!3!(103)!+10!5!(105)!+10!7!(107)!+10!9!(109)!
=10!1!9!+10!3!7!+10!5!5!+10!7!3!+10!9!1!
=10+120+252+120+10
=512
Thus there are 512 subsets with an odd number of elemets of a set with 10 elements.

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