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

We first list all ordered pairs $$(a,b)$$ in $$S\times S$$ that satisfy the restriction $$|a-b|\ge 2$$.

Total ordered pairs in $$S\times S$$ are $$6\times 6 = 36$$.

Forbidden pairs:
• Diagonal pairs with $$a=b$$  → 6 pairs.
• Pairs with $$|a-b|=1$$.
   When $$b=a+1$$: $$(1,2),(2,3),(3,4),(4,5),(5,6)$$ → 5 pairs.
   When $$b=a-1$$: $$(2,1),(3,2),(4,3),(5,4),(6,5)$$ → 5 pairs.
So forbidden pairs = $$6+5+5 = 16$$.

Allowed pairs (with $$|a-b|\ge 2$$) = $$36-16 = 20$$.

Every relation $$R\in X$$ is a subset of these 20 allowed pairs containing exactly 6 elements. Choosing any 6 of the 20 allowed pairs gives such a relation, and every such choice is valid.

Hence $$n(X)=\binom{20}{6}= {}^{m}C_6$$ which implies $$m=20$$.

Final Answer: 20