Question: For n Z+ define Xn = {1, 2, 3, . . . , n}. Given m, n Z+, f: Xm Xn is

For n ∈ Z+ define Xn = {1, 2, 3, . . . , n}. Given m, n ∈ Z+, f: Xm → Xn is called monotone increasing if for all i, j ∈ Xm, 1 < i < j < m => f(i) < f(j). (a) How many monotone increasing functions are there with domain X7 and codomain X5? (b) Answer part (a) for the domain X6 and codomain X9. (c) Generalize the results in parts (a) and (b). (d) Determine the number of monotone increasing functions f: X10 → X8 where f(4) = 4. (e) How many monotone increasing functions f: X7 → X12 satisfy f(5) = 9? (f) Generalize the results in parts (d) and (e).

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