Sync LeetCode submission Runtime - 4 ms (13.43%), Memory - 17.7 MB (62.62%)

Author: jimit105

3216-distribute-candies-among-children-iii/README.md

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+<p>You are given two positive integers <code>n</code> and <code>limit</code>.</p>
+
+<p>Return <em>the <strong>total number</strong> of ways to distribute </em><code>n</code> <em>candies among </em><code>3</code><em> children such that no child gets more than </em><code>limit</code><em> candies.</em></p>
+
+<p>&nbsp;</p>
+<p><strong class="example">Example 1:</strong></p>
+
+<pre>
+<strong>Input:</strong> n = 5, limit = 2
+<strong>Output:</strong> 3
+<strong>Explanation:</strong> 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).
+</pre>
+
+<p><strong class="example">Example 2:</strong></p>
+
+<pre>
+<strong>Input:</strong> n = 3, limit = 3
+<strong>Output:</strong> 10
+<strong>Explanation:</strong> 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).
+</pre>
+
+<p>&nbsp;</p>
+<p><strong>Constraints:</strong></p>
+
+<ul>
+	<li><code>1 &lt;= n &lt;= 10<sup>8</sup></code></li>
+	<li><code>1 &lt;= limit &lt;= 10<sup>8</sup></code></li>
+</ul>

3216-distribute-candies-among-children-iii/solution.py

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+def cal(x):
+    if x < 0:
+        return 0
+    return x * (x - 1) // 2
+
+
+class Solution:
+    def distributeCandies(self, n: int, limit: int) -> int:
+        return (
+            cal(n + 2)
+            - 3 * cal(n - limit + 1)
+            + 3 * cal(n - (limit + 1) * 2 + 2)
+            - cal(n - 3 * (limit + 1) + 2)
+        )