JEE Advance - Mathematics (2024 - Paper 2 Online - No. 14)

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 $A$.

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 $A$.

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 $A$.

If $n(X)={ }^m C_6$, then the value of $m$ is _____

Answer

$$|a-b| \geq 2 \text { or }|b-a|=2$$

Total

$$\begin{array}{lll} a=1 & b=3,4,5,6 & 8 \\ a=2 & b=4,5,6 & 6 \\ a=3 & b=5,6 & 4 \\ a=4 & b=6 & 2 \\ \text { sum }=20 \end{array}$$

$$\begin{aligned} & \mathrm{n}(\mathrm{X})={ }^{20} \mathrm{C}_6={ }^{\mathrm{m}} \mathrm{C}_6 \\ & \mathrm{~m}=20 \end{aligned}$$