[Solved] You are given an unlimited supply of coins of different denominations and a target amount.... Which algorithmic approach guarantees an efficient and correct solution for this problem? | Dynamic Programming: Knapsack

Dynamic Programming: Knapsack - Coin Change II (Count Ways)

You are given an unlimited supply of coins of different denominations and a target amount. You need to find the number of distinct combinations of coins that sum up to the target amount, where the order of coins does not matter. Which algorithmic approach guarantees an efficient and correct solution for this problem?

AGreedy algorithm that picks the largest coin first until the amount is reached

BDynamic programming using a 1D array with nested loops iterating over coins and amounts in increasing order

CBacktracking with pruning to explore all possible permutations of coins

DTop-down recursion without memoization exploring all subsets of coins

Step-by-Step Solution

Solution:

Step 1: Understand problem constraints
The problem requires counting combinations (order does not matter) of unlimited coins summing to amount.
Step 2: Identify suitable algorithm
Greedy fails as it misses combinations; backtracking without memoization is exponential; permutations count order, not combinations. DP with 1D array iterating coins first ensures combinations are counted correctly.
Final Answer:
Option B -> Option B
Quick Check:
DP approach counts combinations efficiently [OK]

Quick Trick: Order coins outer loop to count combinations correctly [OK]

Common Mistakes:

MISTAKES

Using greedy approach
Counting permutations instead of combinations
Not using DP or memoization

Trap Explanation:

PITFALL

Greedy looks efficient but misses valid combinations; permutations count order, inflating results.

Interviewer Note:

CONTEXT

Tests if candidate can recognize the unbounded knapsack pattern for counting combinations.

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More Dynamic Programming: Knapsack Quizzes

Which algorithmic approach guarantees an efficient and correct solution for this problem?

Step 1: Understand problem constraints

Step 2: Identify suitable algorithm

Final Answer:

Quick Check:

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