माना कि S={1,2,3,4,5,6} है, और X, S से S में | vedclass

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$\begin{array}{l}|a-b| \geq 2 	ext { or }|b-a|=2 \ 	ext { Total } \ a =1 \quad b =3,4,5,6 \quad 8 \ a =2 \quad b =4,5,6 \quad 6 \ a =3 \quad b =

Vedclass · Accepted Answer

$20,36$

Vedclass · Answer

$\begin{array}{l}|a-b| \geq 2 	ext { or }|b-a|=2 \ 	ext { Total } \ a =1 \quad b =3,4,5,6 \quad 8 \ a =2 \quad b =4,5,6 \quad 6 \ a =3 \quad b =5,6 \quad 4 \ a =4 \quad b =6 \quad 2 \ \overline{\operatorname{sum}=20} \ n ( X )={ }^{20} C _6={ }^{ m } C _6 \ m =20 \\end{array}$

given $|a-b| \geq 2$ so if

$a=1  b=3,4,5,6  ightarrow  4 	imes 2=8$

$a=1  b=4,5,6 ightarrow  3 	imes 2=6$

$a=1  b=5,6  ightarrow 2 	imes 2=4$

$a=1  b=6  ightarrow  2 	imes 1=2$

$i.e. $Total elements in $X$ is ${ }^{20} C _6$

Now for $n ( Y )$,

range of $R$ has exactly one element $i.e$. second elements must be constant in $R$ and since $R$ must have 6 element so it is not possible to satisfy both condition so $n ( Y )=0$.

$n ( z ) \quad  1 ightarrow 3,4,5,6$

$2 ightarrow 4,5,6$

$3 ightarrow 1,5,6$

$4 ightarrow 1,2,6$

$5 ightarrow 1,2,3$

$6 ightarrow 1,2,3,4$

no. of relation that are function will be

$={ }^4 C _1 	imes{ }^3 C _1 	imes{ }^3 C _1 	imes{ }^3 C _1 	imes{ }^3 C _1 	imes{ }^4 C _1$

$=(4 	imes 3 	imes 3)^2= k ^2$

i.e. $k =36$

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