Coin Change Total Number of Ways in DSA C# - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Coin Change Ways Calculation

What is the output of the following C# code that calculates the total number of ways to make change for 5 using coins {1, 2, 3}?

DSA C#

using System;

class Program {
    static void Main() {
        int[] coins = {1, 2, 3};
        int amount = 5;
        int ways = CountWays(coins, amount);
        Console.WriteLine(ways);
    }

    static int CountWays(int[] coins, int amount) {
        int[] dp = new int[amount + 1];
        dp[0] = 1;
        foreach (int coin in coins) {
            for (int i = coin; i <= amount; i++) {
                dp[i] += dp[i - coin];
            }
        }
        return dp[amount];
    }
}

A 6

B 5

C 7

D 8

💡 Hint

Think about how many combinations of coins 1, 2, and 3 can sum to 5.

The code uses dynamic programming to count ways. For amount 5 and coins {1,2,3}, the total ways are 7.

🧠 Conceptual

intermediate

Understanding DP Array Initialization

In the coin change ways calculation, why is dp[0] initialized to 1?

A Because there is exactly one way to make amount 0: use no coins.

B Because dp[0] stores the total number of coins available.

C Because dp[0] is the base case for amount 1.

D Because dp[0] counts the number of coin types.

💡 Hint

Think about how many ways to make zero amount.

dp[0] = 1 means there is one way to make amount zero: by choosing no coins.

🔧 Debug

advanced

Identify the Error in Coin Change Code

What error will this code produce when run, and why? int[] coins = {1, 2}; int amount = 3; int[] dp = new int[amount]; dp[0] = 1; foreach (int coin in coins) { for (int i = coin; i <= amount; i++) { dp[i] += dp[i - coin]; } } Console.WriteLine(dp[amount]);

A IndexOutOfRangeException because dp array size is amount, but code accesses dp[amount].

B NullReferenceException because dp array is not initialized.

C No error, output is 2.

D Compile-time error due to missing semicolon.

💡 Hint

Check the size of dp array and the indices accessed.

dp array length is amount (3), so valid indices are 0,1,2. Accessing dp[3] causes IndexOutOfRangeException.

🚀 Application

advanced

Number of Ways to Make Change for 10

Using coins {2, 5, 3, 6}, how many ways are there to make change for amount 10?

A 7

B 4

C 6

D 5

💡 Hint

Use dynamic programming or count combinations manually.

The total number of ways to make 10 with coins {2,3,5,6} is 5.

🧠 Conceptual

expert

Why Use Outer Loop Over Coins in Coin Change DP?

In the coin change total ways algorithm, why is the outer loop iterating over coins and the inner loop over amounts?

A To optimize memory usage by limiting dp array size.

B To ensure combinations are counted without duplicates by processing coins in order.

C To count permutations instead of combinations.

D To allow negative coin values in the calculation.

💡 Hint

Think about how order of loops affects counting unique combinations.

Outer loop over coins ensures each coin is considered once per amount, avoiding duplicate counting of same combinations in different orders.

Practice

1. What does the "Coin Change Total Number of Ways" problem ask you to find?

easy

A. The total sum of all coin denominations.

B. The minimum number of coins needed to make a target amount.

C. The maximum value of coins that can be used to reach the target.

D. The total number of ways to make a target amount using given coin denominations.

Solution

1. Step 1: Understand the problem goal

   The problem asks for counting all possible combinations of coins that sum up to the target amount.

2. Step 2: Differentiate from similar problems

   Unlike minimum coins problem, here we count ways, not minimize coins.

3. Final Answer:

   The total number of ways to make a target amount using given coin denominations. -> Option D

4. Quick Check:

   Count ways, not minimum coins [OK]

Hint: Count combinations, not minimum coins [OK]

Common Mistakes:

• Confusing total ways with minimum coins needed
• Thinking about maximum coin value instead of count
• Ignoring that order of coins does not matter

2. Which of the following C# code snippets correctly initializes the DP array for the Coin Change Total Number of Ways problem for amount 5?

easy

A. int[] dp = new int[6]; dp[0] = 1;

B. int[] dp = new int[5]; dp[0] = 1;

C. int[] dp = new int[5]; dp[1] = 1;

D. int[] dp = new int[6]; dp[1] = 1;

Solution

1. Step 1: Determine DP array size

   For amount 5, dp array size should be 6 to include index 0 to 5.

2. Step 2: Initialize base case

   dp[0] = 1 means there is 1 way to make amount 0 (using no coins).

3. Final Answer:

   int[] dp = new int[6]; dp[0] = 1; -> Option A

4. Quick Check:

   Array size = amount+1, dp[0] = 1 [OK]

Hint: DP size = amount + 1; dp[0] = 1 base case [OK]

Common Mistakes:

• Using array size equal to amount instead of amount+1
• Initializing dp[1] instead of dp[0]
• Forgetting to set base case dp[0] = 1

3. What is the output of the following C# code snippet?

int[] coins = {1, 2};
int amount = 3;
int[] dp = new int[amount + 1];
dp[0] = 1;
foreach (int coin in coins) {
  for (int i = coin; i <= amount; i++) {
    dp[i] += dp[i - coin];
  }
}
Console.WriteLine(dp[amount]);

medium

A. 2

B. 3

C. 1

D. 4

Solution

1. Step 1: Initialize dp array

   dp = [1,0,0,0] initially, dp[0]=1 means 1 way to make 0.

2. Step 2: Process coin 1

   For i=1 to 3: dp[1]+=dp[0]=1, dp[2]+=dp[1]=1, dp[3]+=dp[2]=1; dp=[1,1,1,1]

3. Step 3: Process coin 2

   For i=2 to 3: dp[2]+=dp[0]=1+1=2, dp[3]+=dp[1]=1+1=2; dp=[1,1,2,2]

4. Final Answer:

   2 -> Option A

5. Quick Check:

   Ways to make 3 with coins 1,2 = 2 [OK]

Hint: Add dp[i-coin] for each coin and amount [OK]

Common Mistakes:

• Mixing up dp array updates order
• Forgetting to initialize dp[0] = 1
• Miscounting ways by double counting

4. Identify the error in the following C# code for Coin Change Total Number of Ways and select the fix.

int[] coins = {1, 2, 3};
int amount = 4;
int[] dp = new int[amount + 1];
dp[0] = 1;
for (int i = 0; i <= amount; i++) {
  for (int coin : coins) {
    if (i >= coin) {
      dp[i] += dp[i - coin];
    }
  }
}
Console.WriteLine(dp[amount]);

medium

A. Change dp[0] = 1 to dp[1] = 1

B. Change outer loop to 'for (int i = 1; i <= amount; i++)' and inner loop to 'for (int coin in coins)'

C. Replace 'for (int coin : coins)' with 'foreach (int coin in coins)'

D. No error, code is correct

Solution

1. Step 1: Identify incorrect loop syntax

   C# does not support 'for (int coin : coins)', it uses 'foreach (int coin in coins)'.

2. Step 2: Correct the loop syntax

   Replace 'for (int coin : coins)' with 'foreach (int coin in coins)' to fix syntax error.

3. Final Answer:

   Replace 'for (int coin : coins)' with 'foreach (int coin in coins)' -> Option C

4. Quick Check:

   C# uses foreach for collections [OK]

Hint: Use foreach for arrays in C# [OK]

Common Mistakes:

• Using Java-style for-each syntax in C#
• Changing dp base case incorrectly
• Assuming code is correct without checking syntax

5. Given coins = {1, 3, 5} and amount = 7, how many total ways are there to make amount 7? Use the coin change total number of ways logic.

hard

A. 5

B. 4

C. 7

D. 6

Solution

1. Step 1: Initialize dp array

   dp size = 8, dp[0] = 1, dp = [1,0,0,0,0,0,0,0]

2. Step 2: Process coin 1

   Update dp for i=1 to 7: dp[i]+=dp[i-1]; dp becomes [1,1,1,1,1,1,1,1]

3. Step 3: Process coin 3

   For i=3 to 7: dp[3]+=dp[0]=2, dp[4]+=dp[1]=2, dp[5]+=dp[2]=2, dp[6]+=dp[3]=3, dp[7]+=dp[4]=3; dp=[1,1,1,2,2,2,3,3]

4. Step 4: Process coin 5

   For i=5 to 7: dp[5]+=dp[0]=3, dp[6]+=dp[1]=4, dp[7]+=dp[2]=4; dp=[1,1,1,2,2,3,4,4]

5. Final Answer:

   4 -> Option B

6. Quick Check:

   Ways to make 7 with coins 1,3,5 = 4 [OK]

Hint: Update dp for each coin from coin to amount [OK]

Common Mistakes:

• Miscounting ways by skipping coins
• Not updating dp array correctly
• Confusing total ways with minimum coins