Distribute Candies Among Children II | Stars and Bars | Combinatorics | Leetcode 2929

class Solution {
    #define ll long long
    
    ll combinations(ll n){
        if(n<0)
            return 0;
        
        return 1LL*(n+1)*(n+2)/2;
    }
public:
    long long distributeCandies(int n, int limit) {
        ll all_combinations = combinations(n);
        ll one_above_limit_combinations = 3LL*combinations(n-(limit+1));
        ll two_above_limit_combinations = 3LL*combinations(n-2*(limit+1));
        ll three_above_limit_combinations = combinations(n-3*(limit+1));

        ll invalid_combinations =   one_above_limit_combinations -
                                    two_above_limit_combinations +
                                    three_above_limit_combinations;
        
        ll valid_combinations = all_combinations - invalid_combinations;
        return valid_combinations;
    }
};
/*
//JAVA
class Solution {
    private long combinations(long n) {
        if (n < 0) {
            return 0;
        }
        return (n + 1) * (n + 2) / 2;
    }

    public long distributeCandies(int n, int limit) {
        long allCombinations = combinations(n);
        long oneAboveLimitCombinations = 3 * combinations(n - (limit + 1));
        long twoAboveLimitCombinations = 3 * combinations(n - 2 * (limit + 1));
        long threeAboveLimitCombinations = combinations(n - 3 * (limit + 1));

        long invalidCombinations = oneAboveLimitCombinations - twoAboveLimitCombinations + threeAboveLimitCombinations;
        long validCombinations = allCombinations - invalidCombinations;
        return validCombinations;
    }
}

#Python
class Solution:
    def combinations(self, n: int) -> int:
        if n < 0:
            return 0
        return (n + 1) * (n + 2) // 2

    def distributeCandies(self, n: int, limit: int) -> int:
        all_combinations = self.combinations(n)
        one_above_limit_combinations = 3 * self.combinations(n - (limit + 1))
        two_above_limit_combinations = 3 * self.combinations(n - 2 * (limit + 1))
        three_above_limit_combinations = self.combinations(n - 3 * (limit + 1))

        invalid_combinations = (one_above_limit_combinations - 
                               two_above_limit_combinations + 
                               three_above_limit_combinations)
        valid_combinations = all_combinations - invalid_combinations
        return valid_combinations
*/