Given an integer array with all positive numbers and no duplicates, find the number of possible combinations that add up to a positive integer target.

Example:

nums = [1, 2, 3] target = 4

The possible combination ways are:

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7

(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)

Note:

Different sequences are counted as different combinations.

Therefore the output is 7.

Follow up:

• What if negative numbers are allowed in the given array?
• How does it change the problem?
• What limitation we need to add to the question to allow negative numbers?

Solution

状态转移方程：dp[x + y] += dp[x]

其中dp[x]表示生成数字x的所有可能的组合方式的个数。

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class Solution(object):
    def combinationSum4(self, nums, target):
        """
        :type nums: List[int]
        :type target: int
        :rtype: int
        """
        if not nums:
            return 0

        dp = [0]*(target+1)
        dp[0] = 1

        for i in range(target+1):
            for j in nums:
                if i+j <= target:
                    dp[i+j] += dp[i]
        return dp[target]