There are $m$ lids, and $n$ pans. How many possible relations, that describe which lid fits which pan, are there? (We only look at relations, where every pan has at least one fitting lid)
There are $4$ scenarios:
(a) both the lids and pans are distinguishable
(b) only the lids are distinguishable
(c) only the pans are distinguishable
(d) both are indistinguishable
Example: There are $3$ pans and $3$ lids.
Relation $1$: lid $1$ fits pan $1$ and pan $2$, lid $2$ fits pan $3$
Relation $2$: lid $1$ fits pan $1$, lid $2$ fits pan $3$, lid $3$ fits pan $2$
. . . etc.
I am not sure whether a relation like "lid $1$ fits pan $1$ and pan $2$, lid $2$ fits pan $2$ (and not pan $1$), lid $3$ fits pan $3$" is allowed, but would most likely assume, that it is not allowed.