Dynamic Programming: Knapsack - Number of Ways to Make Change
Given the following space-optimized code for counting ways to make change, what is the output when coins = [1, 2, 5] and amount = 5?
Given the following space-optimized code for counting ways to make change, what is the output when coins = [1, 2, 5] and amount = 5?
easy🧾 Code Trace Q3 of Q15
Step-by-Step Solution
Solution:
Step 1: Trace dp array updates
Start dp = [1,0,0,0,0,0]. After coin=1, dp = [1,1,1,1,1,1]. After coin=2, dp = [1,1,2,2,3,3]. After coin=5, dp = [1,1,2,2,3,4].Step 2: Return dp[5]
dp[5] = 4, representing 4 ways to make 5 with coins [1,2,5].Step 3: Re-examine counting
Actually, the number of combinations is 4, but 7 says 7, which is incorrect. The correct answer is 4.Final Answer:
Option A -> Option AQuick Check:
Counted combinations: 5,1; 2,2,1; 1x5; 2,1,1,1 -> 4 ways [OK]
Quick Trick: Trace dp array updates carefully [OK]
Common Mistakes:
MISTAKES- Off-by-one in dp indexing
- Counting permutations instead of combinations
- Mis-updating dp array
Trap Explanation:
PITFALL- Candidates often confuse permutations with combinations, leading to overcounting.
Interviewer Note:
CONTEXT- Tests ability to mentally execute DP code and understand state updates.
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