Problem Statement
Combination Sum IV
Given an array of distinct integers nums and a target integer target, return the number of possible combinations that add up to target.
The test cases are generated so that the answer can fit in a 32-bit integer.
Examples
Example 1:
Input: nums = [1,2,3], target = 4
Output: 7
Explanation: The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)
Note that different sequences are counted as different combinations.
Example 2:
Input: nums = [9], target = 3
Output: 0
Explanation: There are no combinations that sum to target.
Constraints
- 1 <= nums.length <= 200
- 1 <= nums[i] <= 1000
- All the elements of nums are unique
- 1 <= target <= 1000
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- The key insight is to define dp[i] as the number of ways to form the sum i using the given numbers.
- For each number num in nums, if i >= num, then dp[i] += dp[i - num].
- The base case is dp[0] = 1, as there is one way to form the sum 0 (by not selecting any number).
- The problem is similar to the coin change problem, but the order of elements matters here.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Combination Sum IV
Given an array of distinct integers nums and a target integer target, return the number of possible combinations that add up to target.
The test cases are generated so that the answer can fit in a 32-bit integer.
Examples
Example 1:
The possible combination ways are: (1, 1, 1, 1) (1, 1, 2) (1, 2, 1) (1, 3) (2, 1, 1) (2, 2) (3, 1) Note that different sequences are counted as different combinations.
Example 2:
There are no combinations that sum to target.
Problem Breakdown
Constraints:
- 1 <= nums.length <= 200
- 1 <= nums[i] <= 1000
- All the elements of nums are unique
- 1 <= target <= 1000
Key Insights:
This problem can be solved using dynamic programming.
The key insight is to define dp[i] as the number of ways to form the sum i using the given numbers.
For each number num in nums, if i >= num, then dp[i] += dp[i - num].
The base case is dp[0] = 1, as there is one way to form the sum 0 (by not selecting any number).
The problem is similar to the coin change problem, but the order of elements matters here.
Real-World Application
This problem has several practical applications:
Counting Combinations
Counting the number of ways to achieve a target sum using a set of numbers, such as in financial planning or resource allocation.
Game Theory
Calculating the number of possible ways to reach a target score in games with different scoring options.
Permutation Problems
Solving problems that involve counting the number of permutations that satisfy certain conditions.