Let $S = \{1, 2, 3, \dots , 10\}.$ Consider the set $X = \{R : R$ is an equivalence relation on the set S such that R has exactly 42 elements$\}.$ Then the number of elements in X is ____.

$S = \{1, 2, \ldots, 10\}, \quad |S| = 10$ $|R| = 42$ $X = \{R : R \text{ equivalence relation on } S, \ |R| = 42\}$ $\text{Find } |X|$ $R \text{ equivalence} \iff \text{partition of } S$ $|R| = \sum n_i^2 \quad (n_i = \text{class sizes})$ $\sum n_i = 10, \quad \sum n_i^2 = 42$ $\sum n_i^2 - \sum n_i = 42 - 10 = 32$ $\sum n_i(n_i - 1) = 32$ $n_i \ge 7 \Rightarrow n_i(n_i-1) \ge 42 > 32 \ (\text{impossible})$ $n_i = 6: \ 6\cdot 5 = 30 \Rightarrow \text{a part } 2 \Rightarrow \{6,2,1,1\}$ $n_i = 5: \ 5\cdot 4 = 20 \Rightarrow \text{need } 12 = 4\cdot 3 \Rightarrow \{5,4,1\}$ $\text{no other partition works}$ $\{6,2,1,1\}: \ \frac{10!}{6!\,2!\,(1!)^2\,2!} = \frac{3628800}{2880} = 1260$ $\{5,4,1\}: \ \frac{10!}{5!\,4!\,1!} = \frac{3628800}{2880} = 1260$ $|X| = 1260 + 1260 = 2520$