[Solved] Consider the function countWays(amount, coins) that returns the number of ways to make amount using unlimited coins from coins. What is the output of countWays(4, [1, 3, 4])? | Dynamic Programming: Knapsack

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

Consider the function countWays(amount, coins) that returns the number of ways to make amount using unlimited coins from coins. What is the output of countWays(4, [1, 3, 4])?

A4

B4

C2

D5

Step-by-Step Solution

Solution:

Step 1: Identify combinations for amount 4
Possible combinations are: [1,1,1,1], [1,3], [4], and [3,1] is same as [1,3] so not counted separately.
Step 2: Count distinct combinations
There are actually 4 distinct combinations: [1,1,1,1], [1,3], [4], and [1,1,2] is invalid since 2 is not in coins, so ignore. Actually, the combinations are [1,1,1,1], [1,3], [4], and [3,1] is same as [1,3]. So total 3 combinations.
Step 3: Re-examine carefully
Coins are [1,3,4]. Combinations to make 4 are: [1,1,1,1], [1,3], [4]. That's 3 combinations.
Step 4: Check options
4 says 3, 4 says 4. The correct count is 3.
Final Answer:
Option A -> Option A
Quick Check:
Verify combinations manually [OK]

Quick Trick: List combinations systematically to avoid missing any [OK]

Common Mistakes:

MISTAKES

Counting permutations instead of combinations
Missing the combination using coin 4
Double counting combinations with same coins in different order

Trap Explanation:

PITFALL

Counting permutations leads to overcounting; order does not matter here.

Interviewer Note:

CONTEXT

Tests understanding of counting combinations with given coins.

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More Dynamic Programming: Knapsack Quizzes

Consider the function countWays(amount, coins) that returns the number of ways to make amount using unlimited coins from coins. What is the output of countWays(4, [1, 3, 4])?

Step 1: Identify combinations for amount 4

Step 2: Count distinct combinations

Step 3: Re-examine carefully

Step 4: Check options

Final Answer:

Quick Check:

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