Let $S = \{1, 2, 3, \ldots, 10\}$. Consider the set
$$X = \{R : R \text{ is an equivalence relation on the set } S \text{ such that } R \text{ has exactly } 42 \text{ elements}\}.$$
Then the number of elements in $X$ is __________.
Step-by-Step Solution
Key Concept: The number of ordered pairs in an equivalence relation equals $\sum|B_i|^2$ over the equivalence classes. The counting problem reduces to partitioning $n=10$ into positive integers with a fixed sum of squares.
**Step 1: Link $|R|$ to the partition structure**
An equivalence relation on $S$ corresponds to a partition. If the partition has blocks of sizes $n_1, n_2, \ldots, n_k$, then $|R| = \sum n_i^2$. We need $\sum n_i^2 = 42$ with $\sum n_i = 10$.
**Step 2: Find all valid partitions of 10 with $\sum n_i^2 = 42$**
$(6,2,1,1)$: $36+4+1+1=42$. ✓
$(5,4,1)$: $25+16+1=42$. ✓
All other partitions yield a different sum of squares.
**Step 3: Count partitions of type $(6,2,1,1)$**
Choose 6 elements: $\binom{10}{6}=210$. Choose 2 from remaining 4: $\binom{4}{2}=6$. The last 2 are distinct singletons. Total: $210\times6=1260$.
**Step 4: Count partitions of type $(5,4,1)$**
Choose 5 elements: $\binom{10}{5}=252$. Choose 4 from remaining 5: $\binom{5}{4}=5$. Last 1 is determined. Total: $252\times5=1260$.
**Step 5: Total**
$1260+1260 = \mathbf{2520}$.
Correct Answer: