If n is the number of elements in X and m is the number of elements in Y for each we have choices in for its image in a mapping. Therefore we have possible mappings.
Suppose X = {a, b, c} and Y = {d, e}.
1. f: X->Y
f(a) = d
f(b) = d
f(c) = d
2. f: X->Y
f(a) = d
f(b) = d
f(c) = e
3. f: X->Y
f(a) = d
f(b) = e
f(c) = d
4. f: X->Y
f(a) = e
f(b) = d
f(c) = d
5. f: X->Y
f(a) = e
f(b) = e
f(c) = d
6. f: X->Y
f(a) = e
f(b) = d
f(c) = e
7. f: X->Y
f(a) = d
f(b) = e
f(c) = e
8. f: X->Y
f(a) = e
f(b) = e
f(c) = e