Generate N random numbers within a range with a constant sum

Question

I want to generate N random numbers drawn from a specif distribution (e.g uniform random) between [a,b] which sum to a constant C. I have tried a couple of solutions I could

Answer 1

For my answer I'll assume that we have a uniform distribution.

Since we have a uniform distribution, every tuple of C has the same probability to occur. For example for a = 2, b = 2, C = 12, N = 5 we have 15 possible tuples. From them 10 start with 2, 4 start with 3 and 1 starts with 4. This gives the idea of selecting a random number from 1 to 15 in order to choose the first element. From 1 to 10 we select 2, from 11 to 14 we select 3 and for 15 we select 4. Then we continue recursively.

#include 
#include 

std::default_random_engine generator(time(0));
int a = 2, b = 4, n = 5, c = 12, numbers[5];

// Calculate how many combinations of n numbers have sum c
int calc_combinations(int n, int c) {
    if (n == 1) return (c >= a) && (c <= b);
    int sum = 0;
    for (int i = a; i <= b; i++) sum += calc_combinations(n - 1, c - i);
    return sum;
}

// Chooses a random array of n elements having sum c
void choose(int n, int c, int *numbers) {
    if (n == 1) { numbers[0] = c; return; }

    int combinations = calc_combinations(n, c);
    std::uniform_int_distribution distribution(0, combinations - 1);
    int s = distribution(generator);
    int sum = 0;
    for (int i = a; i <= b; i++) {
        if ((sum += calc_combinations(n - 1, c - i)) > s) {
            numbers[0] = i;
            choose(n - 1, c - i, numbers + 1);
            return;
        }
    }
}

int main() { choose(n, c, numbers); }

Possible outcome:

2
2
3
2
3

This algorithm won't scale well for large N because of overflows in the calculation of combinations (unless we use a big integer library), the time needed for this calculation and the need for arbitrarily large random numbers.