Number Of Disjoint Subsets

Probability Level pending

Let $S$ be the set of the first $100$ natural numbers and $A, B$ any two subsets of $S$ .

Compute the number of ordered pairs $\left(A,B\right)$ such that $A \cap B = \emptyset$ .

None of the other answers

3^{100}-2^{100}

3^{100}

2^{100}

\displaystyle{\frac{3^{100}-1}{2}+1}

1 solution

Brian Charlesworth
Sep 28, 2018

For each of the 100 elements there are 3 options, i.e., be placed in subset $A$ , subset $B$ or neither. There are thus $\boxed{3^{100}}$ ordered disjoint pairs $(A,B)$ .

Number Of Disjoint Subsets

1 solution

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