The number of equivalence relations on the set \1,2,3,4\ that contain the ordered pair (1,2) is

An equivalence relation on a set corresponds to a partition of it into classes. Since $(1,2)$ belongs to the relation, the elements $1$ and $2$ must lie in the same class. Treat $\{1,2\}$ as one merged element; the relation then corresponds to a partition of the three objects $\{1,2\},3,4$. The number of partitions of a $3$-element set is the Bell number $B_3=5$: all together, three singletons, and three ways to pair two of them. Hence the answer is $5$. The total number of equivalence relations on four elements is $B_4=15$, which is the wrong choice when the condition is ignored.