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What is the worst-case time complexity of the backtracking solution with pruning for finding all combinations of k distinct numbers from 1 to 9 that sum to n?

medium📊 Complexity Theory Q5 of Q15

Subsets & Combinations - Combination Sum III (K Numbers to N)

What is the worst-case time complexity of the backtracking solution with pruning for finding all combinations of k distinct numbers from 1 to 9 that sum to n?

AO(9^k)

BO(C(9, k) * k)

CO(2^9)

DO(k * n)

Step-by-Step Solution

Solution:

Step 1: Understand search space
We select k numbers from 9 distinct numbers, so combinations count is C(9, k).
Step 2: Analyze complexity
Each combination requires O(k) time to build and verify sums.
Final Answer:
Option B -> Option B
Quick Check:
Combination count times per-combination cost [OK]

Quick Trick: Complexity depends on combinations count C(9,k) times k [OK]

Common Mistakes:

MISTAKES

Assuming exponential 9^k without pruning
Confusing with 2^9 subsets instead of k-sized combinations
Ignoring per-combination construction cost

Trap Explanation:

PITFALL

9^k looks like brute force but pruning limits to combinations C(9,k).

Interviewer Note:

CONTEXT

Tests understanding of combinatorial complexity and pruning impact.

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