Count of Subset Sum
A medium Dynamic Programming problem included in Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Dynamic Programming
- Sheets
- 1
- Core for
- 9 roles
- Platform
- GeeksforGeeks
The problem
Given an array of non-negative integers and a target sum, count the number of subsets whose elements sum to exactly the target.
Example 1
- Input
- nums = [1,1,1,1,1], target = 3
- Output
- 5
- Why
- There are 5 ways to choose a subset of size 3 from five 1s.
Example 2
- Input
- nums = [1,2,3], target = 4
- Output
- 1
- Why
- Only subset [1,3] sums to 4.
Example 3
- Input
- nums = [2,3,5,7], target = 10
- Output
- 1
- Why
- Only subset [3,7] sums to 10.
Constraints
- 1 <= nums.length <= 300
- 1 <= nums[i] <= 100
- 1 <= target <= 1000
How to think about it
Updated 2026-09-09The number of ways to form sum s using a subset of elements is the ways to form s without the current element plus the ways to form s - num with previous elements. Accumulating counts in reverse order across target values prevents using the same element multiple times.
Approaches, worst first
Recursive subset enumeration
time O(2^n) · space O(n)
Recurse through indices with include/exclude choices, returning 1 whenever the remaining target hits 0. Explores 2^n paths, recalculating identical sub-targets.
2D dynamic programming grid
time O(n * target) · space O(n * target)
dp[i][j] holds number of subsets from first i elements summing to j. dp[i][j] = dp[i-1][j] + (j >= nums[i-1] ? dp[i-1][j - nums[i-1]] : 0).
1D backward accumulationWrite this one
time O(n * target) · space O(target)
Array dp of size target + 1 with dp[0] = 1. For each number, iterate j from target down to num: dp[j] += dp[j - num].
Where people lose marks · 3
- Iterating forward from num up to target. Forward iteration allows an element to be added to its own earlier sum, converting subset counts into unbounded multi-set counts.
- Zero elements in the array: if nums contains 0s, each 0 doubles the number of valid subsets because it can be either included or excluded without changing the sum.
- Base case failure: dp[0] must start at 1 to represent the single empty subset summing to 0.
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.
What each operation costs
| Operation | Time |
|---|---|
| fill dynamic programming table of n states | O(n) |
| solve two-dimensional grid of m by n states | O(m * n) |
| space-optimized state transition keeping one row | O(n) |
What usually goes wrong with Dynamic Programming
- Filling a bottom-up table in an order where the current cell needs values that have not been computed yet, reading uninitialized zeros.
- Failing to initialize base cases properly, such as filling a minimization table with zeros instead of infinity, which traps the answer at zero.
- Overwriting values in a 1D space-optimized knapsack array by scanning in the wrong direction, allowing the same item to be chosen multiple times.
Which roles need this problem
Dynamic Programming is a core topic for these 9 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 5 more roles, including Game Developer, Cryptography Engineer, Performance Engineer.
Companies that have asked it
Tags taken from the problem's own GeeksforGeeks page — not a copied list.
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