Two interesting probability problems

First consider question 1:
From the natural number 1,2....N choose m not to repeat the number, which is the maximum expected?
Assuming that the maximum value of a single test is K, then
P (k) =c (k−1m−1) C (MN)
E (Max) =∑K=MNP (k) K=∑K=MNMC (MK) C (MN)
Because ∑K=MNC (MK) =c (m+1n+1)
E (Max) =MC (m+1n+1) C (MN) =m (n+1)/(m+1)
First give the Python code to simulate the validation:

def simulation_notrepeat (n,m):
    tot_average = 0 for
    i in range (10000):
        max = 0
        count = 0
        is_repeated = Set ([]) while
        count < m:
            x = Random.randint (1,n)
            if x to is_repeated:
                is_repeated.add (x)
                Count + = 1
                if x > Max:
                    max = x
        tot_average + max
    print (tot_average/10000)
def test_ Notrepeat (n,m):
    print (m* (n+1)/(m+1))

Question 2: Assuming that question 1 all numbers can be duplicated, then the maximum number of expectations.
Suppose the maximum value of a single test is K
P (k) =km− (k−1) MNM
E (Max) =∑K=1NP (k) k
The specific closed expression I did not solve, there are solutions, please inform the ^_^.
Also using Python validation:

Import Random
def Simulation (n,m):
    tot_average = 0 for
    i in range (10000):
        max = 0 for
        j in Range (m): 
  x = Random.randint (1,n)
            if x > Max:
                max = x
        tot_average = max
    print (tot_average/10000)
def Test_math (n,m):
    average = 0 for
    k in range (1,n+1):
        average = = (K**m-((k-1) **m)) *k
    print (average/( N**M))

def test_notrepeat (n,m):
    print (m* (n+1)/(m+1)
def Test (N,m,simulation,test_math):
    Simulation (N,M)
    Test_math (n,m)
test (100,35,simulation,test_math)
test (100,35,simulation_notrepeat,test_ Notrepeat)

Results:

Expect to see a repeat of the relative repetition of a larger, but very close.