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How many pairs of sets $(A, B)$ are there that satisfy the condition $A, B \subseteq \left\{1, 2,...,5\right\}, A \cap B = \{\}?$

  1. $125$
  2. $127$
  3. $130$
  4. $243$
  5. $257$
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Best answer
55 votes
55 votes

Correct Option: $C$ $243$.

First take $A$ as $\emptyset$ and $B$ as power set of $\{1,2,3,4,5\} $ which is $2^5$. Then take $A$ as set of one element ex: when $A$=$\left \{ 1 \right \}$ then set $B$ could be any of the $2^4$ elements of power set. This will give us $16 \times 5$ . In similar fashion when $A$ consists of $2$ elements we get total pairs $5C2\times 2^3$; when $A$ is of three elements we get $40$; for $4$ elements we get $10$ pairs and when $A$ is of $5$ elements we get one pair which is $A$=$\left \{ 1,2,3,4,5 \right \}$ and $B=\left \{ \right \}.$ So, in total $=32+80+80+40+10+1=243$


Alternative Solution:

for each element in $[n]$, you have 3 choices:

  1. Include it in $A$ but not in $B$
  2. Include it in $B$, but not in $A  $
  3. Include it in neither

so this gives $3^n$ pairs.

 for set $\{1,2,,...,5\}$, $n=5$, $3^5=243$

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