[Solved] You need to find the number of distinct ways to make a target amount using unlimited... Which algorithmic approach guarantees counting all unique combinations without overcounting permutations? | Dynamic Programming: Knapsack

Dynamic Programming: Knapsack - Number of Ways to Make Change

You need to find the number of distinct ways to make a target amount using unlimited supply of given coin denominations. Which algorithmic approach guarantees counting all unique combinations without overcounting permutations?

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

BDynamic programming using a 1D array iterating coins first, then amounts

CBacktracking exploring all permutations of coins to sum to the amount

DSorting coins and using two pointers to find pairs that sum to the amount

Step-by-Step Solution

Solution:

Step 1: Understand problem constraints
The problem requires counting unique combinations, not permutations, of coins to reach the target amount.
Step 2: Identify algorithm that counts combinations without duplicates
Dynamic programming iterating coins first ensures each coin is considered in order, avoiding counting permutations multiple times. The 1D DP approach efficiently accumulates counts for each amount.
Final Answer:
Option B -> Option B
Quick Check:
DP with coin-first iteration counts combinations correctly [OK]

Quick Trick: Coin-first DP avoids permutation overcounting [OK]

Common Mistakes:

MISTAKES

Using greedy misses some combinations
Counting permutations instead of combinations
Using two pointers only works for pair sums

Trap Explanation:

PITFALL

Greedy looks simpler but misses valid combinations; backtracking permutations overcounts; two pointers only find pairs, not all combinations.

Interviewer Note:

CONTEXT

Tests if candidate recognizes correct DP pattern for combination counting.

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More Dynamic Programming: Knapsack Quizzes

Which algorithmic approach guarantees counting all unique combinations without overcounting permutations?

Step 1: Understand problem constraints

Step 2: Identify algorithm that counts combinations without duplicates

Final Answer:

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