Combinatorics
Relations with single range element and functions
MJAT_TS7_P2
Grade 12
Question:
**Paragraph I (continued):** If $n(Y)+n(Z)=k^2$, then $k$ equals:
Step-by-Step Solution
Key Concept: $Y$: relations with exactly 1 element in range and 6 elements in $R$ with $|a-b|\geq 2$. For range $=\{b\}$: all 6 elements of $R$ have form $(a,b)$ with $|a-b|\geq 2$. Count valid $a$'s for each $b$, choose 6. $Z$: functions from $S$ to $S$ with $|f(a)-a|\geq 2$ for all $a$... count via inclusion-exclusion.
$k=\mathbf{36}$.
Correct Answer: 36