Why Generate All Combinations Sum K in DSA Python?

The Big Idea

What if you could instantly find every way to combine numbers to hit your exact target without missing a single one?

The Scenario

Imagine you want to find all ways to pick numbers from a list that add up to a specific total, like choosing coins to make exact change.

The Problem

Trying every possible group by hand or with simple loops is slow and confusing. You might miss some combinations or repeat others, making it hard to get the full list.

The Solution

This method uses a smart step-by-step approach to explore all possible groups without repeating or missing any. It quickly finds all sets that add up to the target number.

Before vs After

✗ Before

numbers = [1, 2, 3]
for i in numbers:
    for j in numbers:
        if i + j == 4:
            print(i, j)

✓ After

def find_combinations(numbers, target, start=0, path=[]):
    if target == 0:
        print(path)
        return
    for i in range(start, len(numbers)):
        if numbers[i] <= target:
            find_combinations(numbers, target - numbers[i], i + 1, path + [numbers[i]])

What It Enables

You can quickly find every unique group of numbers that add up to your target, no matter how big or complex the list is.

Real Life Example

Planning a party budget by finding all combinations of items you can buy without going over your total money.

Key Takeaways

Manual checking misses or repeats combinations easily.

Smart step-by-step search finds all unique groups.

Useful for budgeting, games, and problem solving.

Practice

1. What does the problem "Generate All Combinations Sum K" ask you to find?

easy

A. All unique combinations of numbers that add up to K

B. The largest number less than K

C. The count of numbers equal to K

D. The smallest number greater than K

Solution

1. Step 1: Understand the problem goal

   The problem asks to find all unique sets of numbers that sum exactly to K.

2. Step 2: Clarify uniqueness and sum condition

   Each combination must add up to K without duplicates in the result.

3. Final Answer:

   All unique combinations of numbers that add up to K -> Option A

4. Quick Check:

   Combinations sum to K = All unique combinations of numbers that add up to K [OK]

Hint: Focus on combinations that sum exactly to K [OK]

Common Mistakes:

• Confusing combinations with permutations
• Looking for single numbers equal to K
• Ignoring the sum condition

2. Which of the following Python function signatures is correct to start solving "Generate All Combinations Sum K"?

easy

A. def combinationSumK(nums, k): return int

B. def combination_sum_k(k: int, nums: list[int]) -> list[int]:

C. def combination_sum_k(nums: int, k: int) -> list[int]:

D. def combination_sum_k(nums: list[int], k: int) -> list[list[int]]:

Solution

1. Step 1: Check parameter types and return type

   The function should take a list of integers and an integer k, returning a list of lists of integers.

2. Step 2: Match signature to problem needs

   def combination_sum_k(nums: list[int], k: int) -> list[list[int]]: matches the expected input and output types correctly.

3. Final Answer:

   def combination_sum_k(nums: list[int], k: int) -> list[list[int]]: -> Option D

4. Quick Check:

   Correct signature with list inputs and list of lists output [OK]

Hint: Use list[int] for input and list[list[int]] for output [OK]

Common Mistakes:

• Wrong parameter types
• Incorrect return type
• Swapping parameter order

3. What is the output of the following code?

def comb_sum(nums, k):
    result = []
    def backtrack(start, path, total):
        if total == k:
            result.append(path[:])
            return
        if total > k:
            return
        for i in range(start, len(nums)):
            path.append(nums[i])
            backtrack(i + 1, path, total + nums[i])
            path.pop()
    backtrack(0, [], 0)
    return result

print(comb_sum([2,3,6,7], 7))

medium

A. [[2, 2, 3]]

B. [[2, 3, 6]]

C. [[7]]

D. [[3, 7]]

Solution

1. Step 1: Trace backtracking with input [2,3,6,7] and target 7

   Check combinations starting from index 0, adding numbers without repetition.

2. Step 2: Identify valid combinations summing to 7

   Only [7] sums exactly to 7 without exceeding or repeating indices.

3. Final Answer:

   [[7]] -> Option C

4. Quick Check:

   Only [7] sums to 7 in given list [OK]

Hint: Backtrack with index increment to avoid duplicates [OK]

Common Mistakes:

• Allowing repeated elements
• Not stopping when total > k
• Appending path without copying

4. Find the error in this code snippet for generating combinations summing to k:

def comb_sum(nums, k):
    result = []
    def backtrack(start, path, total):
        if total == k:
            result.append(path[:])
            return
        if total > k:
            return
        for i in range(start, len(nums)):
            path.append(nums[i])
            backtrack(i, path, total + nums[i])
            path.pop()
    backtrack(0, [], 0)
    return result

medium

A. Appending path without copying causes all results to be the same

B. Using backtrack(i, ...) allows repeated elements incorrectly

C. Missing base case for total < k

D. Not initializing result as a global variable

Solution

1. Step 1: Analyze recursive call backtrack(i, ...)

   Using i instead of i+1 allows reuse of the same element multiple times.

2. Step 2: Understand problem requires unique combinations without repeats

   To avoid duplicates, index must increase to i+1 in recursive calls.

3. Final Answer:

   Using backtrack(i, path, total + nums[i]) allows repeated elements incorrectly -> Option B

4. Quick Check:

   Index must increase to avoid repeats [OK]

Hint: Increment index in recursion to prevent repeats [OK]

Common Mistakes:

• Not copying path when appending
• Allowing repeated elements by wrong index
• Ignoring base cases

5. Given the list [1, 2, 3, 4, 5] and target sum 5, which of the following is the correct set of all unique combinations that sum to 5?

hard

A. [[1, 4], [2, 3], [5]]

B. [[1, 2, 2], [5]]

C. [[1, 1, 3], [2, 3]]

D. [[5, 0], [2, 3]]

Solution

1. Step 1: List all combinations of numbers from [1,2,3,4,5] that sum to 5

   Possible sums: 1+4=5, 2+3=5, 5=5.

2. Step 2: Verify uniqueness and validity

   Combinations like [1,2,2] or [1,1,3] are invalid as 2 or 1 repeated without duplicates allowed.

3. Final Answer:

   [[1, 4], [2, 3], [5]] -> Option A

4. Quick Check:

   Unique combos summing to 5 = [[1, 4], [2, 3], [5]] [OK]

Hint: Sum unique sets without repeating elements [OK]

Common Mistakes:

• Including repeated elements not in input
• Adding zero which is not in list
• Ignoring uniqueness of combinations