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Let S = {1, 2, 3, …, 100}. Then number o...

Let S = {1, 2, 3, …, 100}. Then number of non-empty subsets A of S such that the product of elements in A is even is

A

`2^(50)(2^(50)-1)`

B

`2^(50)-1`

C

`2^(50)+1`

D

`2^(100)-1`

Text Solution

Verified by Experts

The correct Answer is:
A
Given, set `S={1,2,3, ..., 100}`.
Total number of non-empty subsets of `'S' =2^(100)-1`
Now, numbers of non-empty subsets of 'S' in which only odd numbers `[1,3,5, ..., 99}`
occurs `=2^(50)-1`
So, the required number of non-empty subsets of 'S' such that product of elements is even.
`=(2^(100)-1)-(2^(50)-1)`
`=2^(100)-2^(50)=2^(50)(2^(50)-1).`
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