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distribute-candies-among-children-ii.cpp
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distribute-candies-among-children-ii.cpp
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// Time: O(1)
// Space: O(1)
// stars and bars, combinatorics, principle of inclusion and exclusion
class Solution {
public:
long long distributeCandies(int n, int limit) {
const auto& nCr = [](int n, int r) { // Time: O(n), Space: O(1)
if (!(0 <= r && r <= n)) {
return static_cast<int64_t>(0);
}
if (n - r < r) {
r = n - r;
}
int64_t result = 1;
for (int k = 1; k <= r; ++k) {
result *= n - k + 1;
result /= k;
}
return result;
};
const auto& nHr = [&](int n, int r) {
return nCr(n + (r - 1), r - 1);
};
static const int R = 3;
int64_t result = 0;
for (int r = 0; r <= R; ++r) {
result += (r % 2 ? -1 : 1) * nCr(R, r) * nHr(n - r * (limit + 1), R);
}
return result;
}
};
// Time: O(n^2)
// Space: O(1)
// optimized brute force
class Solution2 {
public:
long long distributeCandies(int n, int limit) {
int64_t result = 0;
for (int i = max(n - 2 * limit, 0); i <= min(limit, n); ++i) {
result += min(limit, n - i) - max((n - i) - limit, 0) + 1;
}
return result;
}
};