Combinatorics
Relations with exactly 6 elements and difference ≥ 2
MJAT_TS7_P2
Grade 12
Question:
**Paragraph I:** Let $S=\{1,2,3,4,5,6\}$ and $X$ = all relations $R$ from $S$ to $S$ such that: (i) $R$ has exactly 6 elements; (ii) for each $(a,b)\in R$, $|a-b|\geq 2$. Let $Y=\{R\in X:$ range has exactly one element$\}$ and $Z=\{R\in X: R$ is a function$\}$. If $n(X)=\binom{m}{6}$, then $m$ is:
Step-by-Step Solution
Key Concept: Count pairs $(a,b)\in S\times S$ with $|a-b|\geq 2$: exclude $(a,a)$ (6 pairs) and $(a,b)$ with $|a-b|=1$ (10 pairs: $(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(4,5),(5,4),(5,6),(6,5)$). Total eligible pairs $=36-6-10=20$. Choose 6 from 20.
$m=\mathbf{20}$.
Correct Answer: 20