If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $(a, b) \in R$, we have $|a-b| \geq 2$ Let $Y=\{R \in X$ : The…

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If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $(a, b) \in R$, we have $|a-b| \geq 2$ Let $Y=\{R \in X$ : The range of $R$ has exactly one element $\}$ and $Z=\{R \in X: R$ is a function from $S$ to $S\}$. Let $n(A)$ denote the number of elements in a set $A$.

Aicharya · Accepted Answer

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

If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is Let \(S=\{1,2,3,4,5,6\}\) and \(X\) be the set of all…

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