Combination Sum IV
A medium Dynamic Programming interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Dynamic Programming
- Sheets
- 2
- Core for
- 9 roles
The problem
Given an array of distinct positive integers and a target sum, return the number of possible combinations (order matters) that add up to the target using elements from the array.
Example 1
- Input
- nums = [1,3,4], target = 5
- Output
- 6
- Why
- The six ordered ways: 1+1+1+1+1, 1+1+3, 1+3+1, 3+1+1, 1+4 and 4+1.
Example 2
- Input
- nums = [9], target = 3
- Output
- 0
- Why
- No combination of 9 can sum to 3.
Example 3
- Input
- nums = [1,2,3], target = 1
- Output
- 1
- Why
- Only one way: use the single element 1.
Constraints
- 1 <= nums.length <= 200
- 1 <= nums[i] <= 1000
- 1 <= target <= 1000
- All elements in nums are distinct
How to think about it
Updated 2026-09-29Because order matters, (1, 3) and (3, 1) represent distinct sequences. Any sequence summing to t ends with some element x from the array preceded by a sequence summing to t - x. The total ways to reach t is the sum of the ways to reach t - x across all possible choices of x.
Approaches, worst first
Backtracking permutations
time O(n^target) · space O(target)
Recursively subtract numbers from target until reaching 0, counting 1 for every leaf. Branches branch out nums.length times at every level, creating exponential redundant paths.
Target-first dynamic programmingWrite this one
time O(target * n) · space O(target)
Create dp array of size target + 1 with dp[0] = 1. Outer loop runs through target from 1 to target, and inner loop iterates over every number x in nums. If i >= x, add dp[i - x] to dp[i]. Looping target first ensures sequences of different orderings are counted separately.
Where people lose marks · 3
- Inverting loop orders. Looping over numbers on the outside and targets on the inside counts unordered combinations (like Coin Change II) rather than ordered sequences.
- Integer overflow during intermediate additions in typed languages. Although the final answer fits within standard limits for test suites, intermediate states can blow past 32-bit signed integers.
- Missing the base case dp[0] = 1, which represents the single empty sequence needed to initiate sums.
The theory behind it
Dynamic Programming — the ground this problem stands on. All Dynamic Programming problems
What Dynamic Programming is
Dynamic programming is a method for solving a complex problem by breaking it into overlapping subproblems, solving each subproblem only once, and remembering the answers in a lookup table. Instead of recalculating identical questions over and over, future steps look up previous answers directly. By assembling these saved pieces from the bottom up or storing them during recursion, a task that would take billions of steps finishes in a fraction of a second.
When to reach for it
Reach for dynamic programming when questions ask for the maximum profit, minimum cost, total number of distinct ways to achieve a goal, or whether a target can be formed. Signals include overlapping choices where making a choice now affects what choices remain later, but greedy picking fails to find the true global optimum. If drawing a recursive decision tree reveals the same subproblem states repeating across branches, dynamic programming is needed.
How the pattern works
Identify the state variables that uniquely describe a subproblem, such as an array index and remaining capacity. Write the base cases first, representing states whose answers are known without calculation. Next, write the recurrence relation that expresses the current state using previously solved states, taking the minimum, maximum, or sum among your options. Build the solution either top-down by caching recursive returns in a memo table, or bottom-up by filling an array in topological dependency order. When each state depends only on the previous row, compress storage down to a single array.
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