Coin Change Total Number of Ways in DSA C# - Execution Trace

Concept Flow - Coin Change Total Number of Ways

Start with ways array initialized

↓

For each coin

↓

For each amount from coin to target

↓

Update ways[amount

↓

Repeat for all coins

↓

Result: ways[target

We start with an array to count ways. For each coin, we update ways for amounts that can include that coin. Finally, ways[target] gives total ways.

Execution Sample

DSA C#

int[] coins = {1, 2, 3};
int target = 4;
int[] ways = new int[target + 1];
ways[0] = 1;
foreach (var coin in coins)
  for (int amt = coin; amt <= target; amt++)
    ways[amt] += ways[amt - coin];

This code counts how many ways to make amount 4 using coins 1, 2, and 3.

Execution Table

Step	Operation	Amount Updated	ways Array State	Explanation
0	Initialize ways	0	[1, 0, 0, 0, 0]	ways[0] = 1 means 1 way to make amount 0 (no coins)
1	Use coin 1	1	[1, 1, 0, 0, 0]	ways[1] += ways[0] → 1 way to make 1 with coin 1
2	Use coin 1	2	[1, 1, 1, 0, 0]	ways[2] += ways[1] → 1 way to make 2 with coin 1
3	Use coin 1	3	[1, 1, 1, 1, 0]	ways[3] += ways[2] → 1 way to make 3 with coin 1
4	Use coin 1	4	[1, 1, 1, 1, 1]	ways[4] += ways[3] → 1 way to make 4 with coin 1
5	Use coin 2	2	[1, 1, 2, 1, 1]	ways[2] += ways[0] → add ways to make 0 for coin 2
6	Use coin 2	3	[1, 1, 2, 2, 1]	ways[3] += ways[1] → add ways to make 1 for coin 2
7	Use coin 2	4	[1, 1, 2, 2, 3]	ways[4] += ways[2] → add ways to make 2 for coin 2
8	Use coin 3	3	[1, 1, 2, 3, 3]	ways[3] += ways[0] → add ways to make 0 for coin 3
9	Use coin 3	4	[1, 1, 2, 3, 4]	ways[4] += ways[1] → add ways to make 1 for coin 3
10	End	-	[1, 1, 2, 3, 4]	All coins processed, ways[4] = 4 total ways

💡 All coins processed, ways array fully updated, ways[target] = 4

Variable Tracker

Variable	Start	After coin 1	After coin 2	After coin 3	Final
ways	[1,0,0,0,0]	[1,1,1,1,1]	[1,1,2,2,3]	[1,1,2,3,4]	[1,1,2,3,4]
coin	N/A	1	2	3	3
amt	N/A	1 to 4	2 to 4	3 to 4	N/A

Key Moments - 3 Insights

Why do we set ways[0] = 1 at the start?

ways[0] = 1 means there is exactly one way to make amount 0: by choosing no coins. This is the base case that allows building up ways for higher amounts, as seen in execution_table step 0.

Why do we add ways[amt - coin] to ways[amt] inside the loop?

Adding ways[amt - coin] means we count all ways to make the smaller amount (amt - coin) and then add the current coin to reach amt. This accumulates all combinations, shown in steps 1-9.

Why do we loop coins first, then amounts, not the other way?

Looping coins first ensures combinations are counted without duplicates. If we looped amounts first, we might count the same combination multiple times. The execution_table shows this order clearly.

Visual Quiz - 3 Questions

Look at the execution_table at step 7, what is ways[4] after updating with coin 2?

A 1

B 2

C 3

D 4

💡 Hint

Check the 'ways Array State' column at step 7 in execution_table

At which step does ways[3] first become 3?

A Step 8

B Step 7

C Step 4

D Step 9

💡 Hint

Look at ways[3] values in 'ways Array State' column across steps

If we remove coin 1 from coins, what would ways[1] be after processing all coins?

A 2

B 0

C 1

D 3

💡 Hint

Without coin 1, no coin can make amount 1, check variable_tracker for ways after coin 1

Concept Snapshot

Coin Change Total Number of Ways:
- Initialize ways array with ways[0] = 1
- For each coin, update ways[amount] += ways[amount - coin]
- Loop coins outer, amounts inner to avoid duplicates
- ways[target] holds total combinations
- Dynamic programming builds solutions from smaller amounts

Full Transcript

This visualization shows how to count the total number of ways to make a target amount using given coin denominations. We start with an array 'ways' where ways[0] = 1, meaning one way to make zero amount. For each coin, we update the ways array for amounts from the coin value up to the target. We add ways[amount - coin] to ways[amount], accumulating all combinations that include the current coin. The execution table traces each update step by step, showing how the ways array changes. The variable tracker summarizes the array state after processing each coin. Key moments clarify why ways[0] is 1, why we add ways[amount - coin], and why the coin loop is outer. The quiz tests understanding of specific steps and the effect of removing coins. The concept snapshot summarizes the approach as a dynamic programming solution building from smaller amounts to the target.

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