P&C - 11 (IIFT-2010)

How many subsets of {1,2,3,…11} contain at least one even integer?

Solution follows here:

Solution:

{1,2,3,…11}

Odd integers – 1,4,5,7,9,11 – six in number

Even integers – 2,4,6,8,10 – five in number

This is based on Permutations & Combinations- theory of all possible selections. Check out the following link on this blog for concept:

http://mathbyvemuri.blogspot.com/2011/10/permutations-combinations-all-possible.html

Number of all possible selections from the given ‘n’ things = 2ⁿ-1

Number of all possible sets from the given 11 numbers = 2¹¹-1

Number of all possible sets from the given 6 odd-numbers = 2⁶-1

∴ Number of all possible sets including at least one even number

= (Number of all possible sets from all 11 numbers) – (Number of all possible sets from the 6 odd-numbers)

= (2¹¹-1) – (2⁶-1) = 2048-64 = 1984

Answer (3)