Question

The number of unordered pairs $(A,B)$ of subsets of the sets $S=\{1,2,3,4,5,6\}$ such that $A\cap B=\varphi$ and $A\cup B=S$ is

• A. $32$
• B. $64$
• C. $128$
• D. $63$

Answer 1

The correct option is A $32$

$A\cup B=S$ i.e. if set $A$ contains $x$ elements then set $B$ must contain $6-x$ elements.
Now let us hold the subsets starting from selecting $0$ elements for $A,6$ for $B;1$ element for $A,5$ for $B;\dots \dots$

$\therefore \text{Required number of subsets}={}^6{C}_0\cdot 1+{}^6{C}_1\cdot 1+{}^6{C}_2\cdot 1+\dots +{}^6{C}_6=2^6$

$\therefore$ Unordered pairs of subsets $=\frac{2^6}{2}=2^5=32$