Let X denote the number of elements in set X . Let S = 1 ,   2 ,   3 ,   4 ,   5 , &#160…

Since $A$ and $B$ are independent events
$P\left(A\cap B\right)=P\left(A\right)P\left(B\right)$
$\frac{\left|A\cap B\right|}{\left|S\right|}=\frac{\left|A\right|}{\left|S\right|}·\frac{\left|B\right|}{\left|S\right|}$
$\begin{array}{cc}\left|S\right| \left|A\cap B\right|=\left|A\right|·\left|B\right|& \left(\therefore \left|x\right|=\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{ }\mathrm{i}\mathrm{n}\mathrm{ }\mathrm{s}\mathrm{e}\mathrm{t}\right)\end{array}$
$\left|S\right|=6$
$6\left|A\cap B\right|=\left|A\right|\left|B\right|$
$\begin{array}{cc}6n\left(A\cap B\right)=n\left(A\right)n\left(B\right)& \left(\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{ }\left|x\right|=n\left(x\right)\right)\end{array}$
Case $\mathrm{I}:$
$n\left(A\cap B\right)=1$
$\Rightarrow n\left(A\right)n\left(B\right)=6$
$\left(\mathrm{a}\right) \begin{array}{cc}n\left(A\right)=6, n\left(B\right)=1& \mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{w}\mathrm{a}\mathrm{y}\mathrm{s}={}_{\mathrm{ }}{}^{6}C_6\times {}_{\mathrm{ }}{}^{6}C_1=6\end{array}$

$\left(b\right) n\left(A\right)=3, n\left(B\right)=1 \mathrm{Number} \mathrm{of} \mathrm{ways}={}^{6}C_3\times {}^{3}C_1\times {}^{5}C_2=180$

Case $\mathrm{I}\mathrm{I}:$
$n\left(A\cap B\right)=2$
$\Rightarrow n\left(A\right)n\left(B\right)=12$
$\begin{array}{ccc}\left(a\right) n\left(A\right)=6& n\left(B\right)=2& \mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{w}\mathrm{a}\mathrm{y}\mathrm{s}\end{array}={}_{ }{}^6{C}_6{}_{ }{}^6{C}_2=15$
$\begin{array}{ccc}\left(b\right) n\left(A\right)=4& n\left(B\right)=3& \mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{w}\mathrm{a}\mathrm{y}\mathrm{s}=\end{array}{}_{ }{}^{6}C_4\times {}_{ }{}^{4}C_2\times {}_{ }{}^{2}C_1=180$
Case $\mathrm{I}\mathrm{I}\mathrm{I}:$
$n\left(A\cap B\right)=3$
$\Rightarrow n\left(A\right)n\left(B\right)=18$
$\left(a\right) \begin{array}{ccc}n\left(A\right)=6,& n\left(B\right)=3 \mathrm{Number}\mathrm{ }\mathrm{of}\mathrm{ }\mathrm{ways}\Rightarrow & {}_{ }{}^{6}C_6\times {}_{ }{}^{6}C_3=20\end{array}$
Case $\mathrm{I}\mathrm{V}:$
$n\left(A\cap B\right)=4$
$\Rightarrow n\left(A\right)n\left(B\right)=24$
$\begin{array}{ccc}\left(a\right) n\left(A\right)=6& n\left(B\right)=4 \mathrm{Number}\mathrm{ }\mathrm{of}\mathrm{ }\mathrm{ways}\Rightarrow & {}_{ }{}^{6}C_6\times {}_{ }{}^{6}C_4=15\end{array}$
Case $\mathrm{V}:$
$n\left(A\cap B\right)=5$
$\Rightarrow 1n\left(A\right)n\left(B\right)=30$
$\begin{array}{ccc}\left(a\right) n\left(A\right)=6& n\left(B\right)=5 \mathrm{Number}\mathrm{ }\mathrm{of}\mathrm{ }\mathrm{ways}\Rightarrow & {}_{ }{}^{6}C_6{}_{ }{}^{6}C_5=6\end{array}$
Total ordered pairs $=422$