How many subsets of a set with $100$ elements have more than $2$ element?

Question

How many subsets of a set with $$100$$ elements have more than $$2$$ element?

Approach

Number of subsets of a set with $$100$$ elements =$$2^{100}$$

Number of subsets of a set with $$100$$ elements having more than $$2$$ element

=$$2^{100}$$-Number of subsets of a set with $$100$$ elements having less than $$2$$ element$$(X)$$

$$X$$=Number of subsets of a set with $$100$$ elements having no element $$( \phi)$$+ Number of subsets of a set with $$100$$ elements having one element =1+100

Hence,

Number of subsets of a set with $$100$$ elements having more than $$2$$ element=

$$2^{100}-101$$

Am I right?

1 Answer

You are wrong. You forgot to include sets with exactly two elements.

The correct formula would be

Number of subsets = Number of sets with more than 2 elements + number of sets with 2 elements + number of sets with less than 2 elements.

• will the answer be $2^{100}-101-\binom{100}{2}$ =$2^{100}-101-4950$ ? isn't it? – laura May 11 '17 at 12:27
• @laura Yup, that's the answer! – 5xum May 11 '17 at 12:28