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) ∈ R, we have |a − b| ≥ 2.

Let Y = {R ∈ X : The range of R has exactly one element } and Z = {R ∈ X : R is a function from S to S}.

Let n(A) denote the number of elements in a set A.

If n(X) = ^mC₆, then the value of m is _____

The correct answer is 20.

Step 1: Understand the given set and conditions

We are given the set $S=\{1,2,3,4,5,6\}$.

A relation $R$ from $S$ to $S$ is a subset of the Cartesian product $S\times S$.

The set $X$ consists of all relations $R$ from $S$ to $S$ that satisfy the following two conditions:

1. $R$ has exactly 6 elements, i.e., $\left|R\right|=6$.
2. For each ordered pair $(a,b)\in R$, we have $|a-b|\ge 2$.

Step 2: Determine all valid ordered pairs $(a,b)\in S\times S$ satisfying $|a-b|\ge 2$

Let us count all possible ordered pairs $(a,b)$ from $S\times S$ where the absolute difference between $a$ and $b$ is at least 2:

For $a=1$: $b\in \{3,4,5,6\}$ (4 pairs)
For $a=2$: $b\in \{4,5,6\}$ (3 pairs)
For $a=3$: $b\in \{1,5,6\}$ (3 pairs)
For $a=4$: $b\in \{1,2,6\}$ (3 pairs)
For $a=5$: $b\in \{1,2,3\}$ (3 pairs)
For $a=6$: $b\in \{1,2,3,4\}$ (4 pairs)

Total number of valid pairs = $4+3+3+3+3+4=20$.

Step 3: Calculate the number of elements in set $X$

Since any relation $R\in X$ is formed by selecting exactly 6 distinct pairs out of these 20 valid pairs, the total number of such relations is:

$n\left(X\right)={}^{20}{C}_6$

Step 4: Find the value of $m$

We are given that $n\left(X\right)={}^m{C}_6$.

Comparing ${}^m{C}_6={}^{20}{C}_6$, we get:

$m=20$