Question

WILL GIVE BRANLIEST.

Let S = {1, 2, 3, ..., 12}. How many subsets of S, excluding the empty set, have an even
sum but not an even product?

Answer 1

Given : S = {1, 2, 3, ..., 12}

To find : How many subsets of S, excluding the empty set, have an even

sum but not an even product

Solution:

an even  sum but not an even product

Hence no even number would be in set

So we left with odd numbers only

1 , 3 , 5  , 7  ,  9  , 11

now Sets with even number of elements will have sum even

Set with odd number elements will have sum odd

So Sets with 2 elements , 4 elements , 6 elements

and available number of elements are  6  ( 1 , 3 , 5  , 7  ,  9  , 11 )

Number of Sets with 2 elements   =  ⁶C₂  = 15

Number of Sets with 4 elements   =  ⁶C₄  = 15

Number of Sets with 6 elements   =  ⁶C₆  = 1

15 + 15 + 1 = 31

Hence subsets of S excluding the empty set, having an even  sum but not an even product = 31

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