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

The correct answer is 20.

To find the value of $m$ in the given equation, we need to analyze the underlying mathematical context of this problem, which is typically defined as follows:

Let $S=\{1,2,3,4,5,6\}$ be a set. Let $X$ be the set of all relations $R$ from $S$ to $S$ such that:
1. $R$ has exactly 6 elements (ordered pairs).
2. For each ordered pair $(a,b)\in R$, the condition $|a-b|\ge 2$ is satisfied.

We want to find $n\left(X\right)$, which is the total number of such relations.

Step 1: Determine the total number of ordered pairs in $S\times S$
The number of elements in set $S$ is 6. Thus, the total number of ordered pairs in the Cartesian product $S\times S$ is:
$6\times 6=36$

Step 2: Count the ordered pairs $(a,b)$ that do NOT satisfy the condition $|a-b|\ge 2$
The pairs that do not satisfy the condition are those where:
$|a-b|<2$
This means either $|a-b|=0$ or $|a-b|=1$.

Case 2.1: $|a-b|=0$ (i.e., $a=b$)
These are the diagonal elements. There are 6 such pairs:
$(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)$
Total = 6 pairs.

Case 2.2: $|a-b|=1$
These are the pairs of consecutive elements: - If $a<b$, the pairs are: $(1,2),(2,3),(3,4),(4,5),(5,6)$ (5 pairs)
- If $a>b$, the pairs are: $(2,1),(3,2),(4,3),(5,4),(6,5)$ (5 pairs)
Total = $5+5=10$ pairs.

Summing the excluded pairs:
$6+10=16\text{pairs}$

Step 3: Calculate the number of allowed pairs
Subtracting the excluded pairs from the total pairs:
$36-16=20\text{allowed pairs}$

Step 4: Find the number of relations in $X$
A relation $R$ in $X$ is formed by choosing exactly 6 ordered pairs from the 20 allowed pairs. Therefore, the number of such relations is given by the combination formula:
$n\left(X\right)={}^{20}C_6$

Step 5: Determine $m$
Comparing the expression to the given equation $n\left(X\right)={}^{m}C_6$, we have:
${}^{m}C_6={}^{20}C_6$
This directly gives:
$m=20$