solve problem

leet-code/2928-distribute-candies-among-children-1/README.md

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+# 2928-distribute-candies-among-children-1
+
+You are given two positive integers n and limit.
+
+Return the total number of ways to distribute n candies among 3 children such that no child gets more than limit candies.
+
+Example 1:
+
+Input: n = 5, limit = 2
+Output: 3
+Explanation: There are 3 ways to distribute 5 candies such that no child gets more than 2 candies: (1, 2, 2), (2, 1, 2) and (2, 2, 1).
+Example 2:
+
+Input: n = 3, limit = 3
+Output: 10
+Explanation: There are 10 ways to distribute 3 candies such that no child gets more than 3 candies: (0, 0, 3), (0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0) and (3, 0, 0).
+
+Constraints:
+
+1 <= n <= 50
+1 <= limit <= 50

leet-code/2928-distribute-candies-among-children-1/index.js

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+/**
+ * @param {number} n
+ * @param {number} limit
+ * @description |A ∪ B ∪ C| = |A| + |B| + |C|
+             - |A ∩ B| - |A ∩ C| - |B ∩ C|
+             + |A ∩ B ∩ C|
+ * @return {number}
+ */
+var distributeCandies = function(n, limit) {
+    const numberOfAllCombinations = nChooseK(n + 2, 2)
+    const numberOfDistinctChildren = 3
+    const numberOfCombinationsSuchThatOneChildExceedsLimit = n - limit + 1 >= 0
+        ? numberOfDistinctChildren * nChooseK(n - limit + 1, 2)
+        : 0
+    const numberOfDistinctPairsOfChildren = 3
+    const numberOfCombinationsSuchThatTwoChildrenExceedsLimit = n - limit *2 >= 0
+        ? numberOfDistinctPairsOfChildren * nChooseK(n - limit * 2, 2)
+        : 0
+    const numberOfCombinationsSuchThatThreeChildrenExceedsLimit = n - limit * 3 -1 >= 0
+        ? nChooseK(n - limit * 3 - 1, 2)
+        : 0
+
+    return numberOfAllCombinations
+        - numberOfCombinationsSuchThatOneChildExceedsLimit
+        + numberOfCombinationsSuchThatTwoChildrenExceedsLimit
+        - numberOfCombinationsSuchThatThreeChildrenExceedsLimit
+};
+
+function nChooseK(n, k) {
+  if (k < 0 || k > n) return 0;
+  if (k === 0 || k === n) return 1;
+  k = Math.min(k, n - k);
+  let result = 1;
+  for (let i = 1; i <= k; i++) {
+    result *= n--;
+    result /= i;
+  }
+  return result;
+}