Coin Change Total Number of Ways in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Coin Change Total Number of Ways problem?

To find how many different ways we can make a target amount using given coin denominations, where each coin can be used unlimited times.

beginner

In the Coin Change problem, what does the dynamic programming table represent?

Each cell represents the number of ways to make a certain amount using coins up to a certain index.

intermediate

Why do we add dp[i][j - coin] to dp[i-1][j] in the DP solution?

dp[i-1][j] counts ways without current coin; dp[i][j - coin] counts ways including current coin at least once.

beginner

What is the base case for the Coin Change DP solution?

There is exactly 1 way to make amount 0: by choosing no coins.

intermediate

How does the order of loops affect the Coin Change Total Number of Ways solution?

Looping coins first then amounts ensures combinations are counted without duplicates.

What does dp[j] represent in a 1D DP array for Coin Change?

A Number of coins used to make amount j

B Number of ways to make amount j

C Minimum coins needed for amount j

D Maximum coins used for amount j

dp[j] stores the total number of ways to make amount j using the given coins.

Which of these is a valid coin denomination array for the problem?

A [1, 2, 5]

B [-1, 2, 3]

C [0, 1, 2]

D [1, 2, 2]

Coin denominations must be positive integers and unique; negative or zero values are invalid.

If the amount is 0, how many ways are there to make change?

A 1

B 0

C Amount of coins

D Depends on coin denominations

There is exactly 1 way to make amount 0: by choosing no coins.

In the DP solution, why do we iterate coins before amounts?

A To minimize coins used

B To count permutations

C To count combinations without duplicates

D To maximize coins used

Iterating coins first ensures combinations are counted without duplicates.

What is the time complexity of the Coin Change Total Number of Ways DP solution?

A O(n + amount)

B O(amount^2)

C O(n^2)

D O(n * amount)

The solution uses nested loops over coins (n) and amounts, resulting in O(n * amount) time.

Explain how dynamic programming helps solve the Coin Change Total Number of Ways problem.

Think about how you can reuse answers for smaller amounts to find answers for bigger amounts.

Describe the difference between counting combinations and permutations in the Coin Change problem and how loop order affects it.

Consider how changing the order of loops changes what you count.

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