Let $|X|$ denote the number of elements in a set $X$. Let $S = \{1, 2, 3, 4, 5, 6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \le |B| < |A|$, equals ____.

$S=\{1,2,3,4,5,6\}, \quad |S|=6$

$A,\ B \subseteq S, \quad A \text{ and } B \text{ independent}$

$\text{To find: number of ordered pairs } (A,B)$

$\text{with } 1\le |B|<|A|$

$P(A\cap B)=P(A)\,P(B)$

$\dfrac{|A\cap B|}{6}=\dfrac{|A|}{6}\cdot\dfrac{|B|}{6}$

$\Rightarrow |A\cap B|=\dfrac{|A|\,|B|}{6}$

$\text{Let } |A|=a,\ |B|=b,\ c:=|A\cap B|=\dfrac{ab}{6}$

$1\le b<a\le 6,\quad 6\mid ab$

$\text{Plan: } 1.\ \text{list } (a,b):\ 1\le b<a\le 6,\ 6\mid ab$

$2.\ \#(a,b)=\binom{6}{a}\binom{a}{c}\binom{6-a}{b-c}$

$3.\ \text{sum over all valid } (a,b)$

$\text{choose } A:\ \binom{6}{a}$

$\text{choose } A\cap B\subseteq A:\ \binom{a}{c}$

$\text{choose } B\setminus A\subseteq S\setminus A:\ \binom{6-a}{b-c}$

$6\mid ab:\ (a,b)=(3,2),\ (4,3)$

$a=6:\ b=1,2,3,4,5$

$a=2,\,5:\ \text{none}$

$(a,b)=(3,2):\ c=1$

$\binom{6}{3}\binom{3}{1}\binom{3}{1}=20\cdot 3\cdot 3=180$

$(a,b)=(4,3):\ c=2$

$\binom{6}{4}\binom{4}{2}\binom{2}{1}=15\cdot 6\cdot 2=180$

$a=6:\ c=b,\ B\subseteq A$

$\text{count}=\binom{6}{b},\quad b=1,\dots,5$

$\binom{6}{1}+\binom{6}{2}+\binom{6}{3}+\binom{6}{4}+\binom{6}{5}$

$=6+15+20+15+6=62$

$\text{Total}=180+180+62$

$\therefore\ \text{number of ordered pairs }=422$