{ "data": { "question": { "questionId": "3201", "questionFrontendId": "2929", "boundTopicId": null, "title": "Distribute Candies Among Children II", "titleSlug": "distribute-candies-among-children-ii", "content": "
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.
\n
Example 1:
\n\n\nInput: n = 5, limit = 2\nOutput: 3\nExplanation: 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).\n\n\n
Example 2:
\n\n\nInput: n = 3, limit = 3\nOutput: 10\nExplanation: 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).\n\n\n
\n
Constraints:
\n\n1 <= n <= 106
1 <= limit <= 106
i
which means 0 <= i <= min(limit, n)
.",
"Suppose the 2nd child gets j
candies. Then 0 <= j <= limit
and i + j <= n
.",
"The 3rd child will hence get n - i - j
candies and we should have 0 <= n - i - j <= limit
.",
"After some transformations, for each i
, we have max(0, n - i - limit) <= j <= min(limit, n - i)
, each j
corresponding to a solution.\r\nSo the number of solutions for some i
is max(min(limit, n - i) - max(0, n - i - limit) + 1, 0)
. Sum the expression for every i
in [0, min(n, limit)]
."
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