Generate a list of whole numbers uniformly at random with a given sum

I have seen several posts on this subject already, however they all seem unnecessarily complicated, or wrong --- the following proposal does not suffer from the former problem (it is simple), but possibly the latter (that it is wrong).

My goal is to generate s whole numbers, i.e., positive integers, uniformly at random, such that their sum is n. To me, the following solution of generating n random numbers between 1 and s, and then outputting the frequencies gets what we want:

import random
from collections import defaultdict

samples = list()

for i in range(n) :
    samples.append(random.randint(1,s))

hist = defaultdict(int)

for sample in samples :
    hist[sample] += 1

freq = list()

for j in range(s) :
    freq.append(hist[j+1])

print('list:', freq)
print('sum:', sum(freq))

So, for example, if we wanted s=10 random whole numbers which sum up to n=100, we would get from this procedure, for example

list: [11, 7, 9, 12, 16, 13, 9, 10, 8, 5]
sum: 100

Since I am no statistician by any means, I fear that this generates numbers which are not truly uniformly distributed. Any comments/analysis would be greatly appreciated