Question

If \( n(X) = \binom{m}{6} \), then the value of \( m \) is _____.

Hint:

To count relations, ensure each pair satisfies the given conditions and compute the total combinations.

Answer 1

To satisfy the condition \( |a - b| \geq 2 \), \( b \) must be at least 2 units away from \( a \). The possible pairs are:

\[ \begin{array}{|c|l|} \hline a & b \\ \hline 1 & 3, 4, 5, 6 \\ 2 & 4, 5, 6 \\ 3 & 1, 5, 6 \\ 4 & 1, 2, 6 \\ 5 & 1, 2, 3 \\ 6 & 1, 2, 3, 4 \\ \hline \end{array} \]

Counting all valid pairs:

\(\text{Total pairs} = 20.\)

Thus:

\[ n(X) = \binom{20}{6} \quad \Rightarrow \quad m = 20. \]