Consider Sm = {1,2,3,...,m} and Sn = {1,2,3,...,n} with m,n ≥ 3.
(a) How many functions f are there from Sm to Sn such that f(x) = 1 for at least one x ∈ Sm?
(b) How many functions f are there from Sm to Sn such that f(x) ∈{1,2} for at least one x ∈ Sm?
(c) How many functions f are there from Sm to Sn such that f(x) = 1 for at least one x ∈ Sm and f(y) =/= 2 for any y ∈ Sm?
(d) How many functions f are there from Sm to Sn such that f(x) = 1 for at least one x ∈ Sm and f(y) = 2 for at least one y ∈ Sm?