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 order does not matter.

beginner

In the Coin Change problem, why do we use dynamic programming?

Because it helps us build solutions for smaller amounts step-by-step and reuse those results to find the total ways for the target amount efficiently.

beginner

What does the dp array represent in the Coin Change Total Number of Ways solution?

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

intermediate

How do we update the dp array when processing each coin?

For each coin, we add dp[j - coin] to dp[j] for all j from coin to target amount, meaning we add ways that include the current coin.

intermediate

Why does the order of loops matter in the Coin Change Total Number of Ways problem?

Using coins in the outer loop and amounts in the inner loop ensures combinations are counted without duplicates, as it fixes coin order and avoids permutations.

What does dp[0] represent in the Coin Change problem?

A Total amount of coins

B Number of coins used

C Number of ways to make amount 0, which is 1

D Zero ways to make amount 0

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

If coins = [1, 2] and amount = 3, how many ways are there to make 3?

A 2

B 3

C 1

D 4

Ways: (1+1+1), (1+2), (2+1) but order does not matter, so (1+2) and (2+1) count as one. Total 2 ways.

Which loop order correctly counts combinations without duplicates?

A Loop order does not matter

B Amounts outer loop, coins inner loop

C Both orders count duplicates

D Coins outer loop, amounts inner loop

Coins outer loop ensures combinations are counted once by fixing coin order.

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

A O(n * amount)

B O(amount^2)

C O(n^2)

D O(n + amount)

Where n is number of coins and amount is the target sum.

What initial value should dp array have before processing coins?

A All negative ones

B All zeros except dp[0] = 1

C All zeros

D All ones

dp[0] = 1 because there is one way to make amount 0; others start at 0.

Explain how the dynamic programming array is built to solve the Coin Change Total Number of Ways problem.

Think about how smaller amounts help build solutions for bigger amounts.

Describe why the order of loops matters when counting the total number of ways to make change.

Consider how fixing coin order prevents counting the same combination multiple times.

Practice

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

easy

A. The total number of ways to make a target amount using given 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 sum of all coin denominations.

Solution

1. Step 1: Understand the problem goal

   The problem asks for counting all possible ways to combine coins to reach a target amount.

2. Step 2: Differentiate from similar problems

   Unlike minimum coins problem, this counts combinations, not minimum coins.

3. Final Answer:

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

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 it finds maximum coin value
• Assuming it sums coin denominations

2. Which of the following is the correct way to declare the DP array for the Coin Change Total Number of Ways problem in C?

easy

A. int dp[target];

B. int dp[target+1];

C. int dp[target+1][coins];

D. int dp[coins+1];

Solution

1. Step 1: Understand DP array size

   We need to store ways for amounts from 0 to target, so size is target+1.

2. Step 2: Choose correct declaration

   One-dimensional array of size target+1 is used to store ways for each amount.

3. Final Answer:

   int dp[target+1]; -> Option B

4. Quick Check:

   Array size = target+1 [OK]

Hint: DP array size must cover 0 to target inclusive [OK]

Common Mistakes:

• Using dp[target] misses amount 0
• Using 2D array unnecessarily
• Using dp[coins+1] which is unrelated

3. What is the output of the following C code snippet for coins = {1, 2} and target = 3?

int coins[] = {1, 2};
int target = 3;
int dp[4] = {1, 0, 0, 0};
for (int i = 0; i < 2; i++) {
    for (int j = coins[i]; j <= target; j++) {
        dp[j] += dp[j - coins[i]];
    }
}
printf("%d", dp[target]);

medium

A. 1

B. 3

C. 2

D. 4

Solution

1. Step 1: Initialize dp array

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

2. Step 2: Process coin 1

   For j=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 j=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 C

5. Quick Check:

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

Hint: Add ways from smaller amounts using each coin [OK]

Common Mistakes:

• Not initializing dp[0] to 1
• Updating dp array incorrectly
• Mixing order of loops

4. Identify the error in this code snippet for calculating total ways to make change:

int coins[] = {1, 2, 3};
int target = 4;
int dp[5] = {0};
dp[0] = 1;
for (int i = 0; i <= 3; i++) {
    for (int j = coins[i]; j <= target; j++) {
        dp[j] += dp[j - coins[i]];
    }
}
printf("%d", dp[target]);

medium

A. Loop i should run from 0 to 2, not 0 to 3.

B. dp array size is too small.

C. dp[0] should be initialized to 0.

D. Inner loop should start from 0, not coins[i].

Solution

1. Step 1: Check loop bounds for coins array

   coins has 3 elements indexed 0 to 2, but loop runs i=0 to 3 (4 iterations) causing out-of-bounds access.

2. Step 2: Correct loop range

   Change loop to i=0 to 2 to avoid accessing coins[3] which is invalid.

3. Final Answer:

   Loop i should run from 0 to 2, not 0 to 3. -> Option A

4. Quick Check:

   Array index out-of-bounds error [OK]

Hint: Loop over coins must match array length exactly [OK]

Common Mistakes:

• Using wrong loop bounds causing crashes
• Incorrect dp initialization
• Starting inner loop incorrectly

5. Given coins = {1, 3, 5} and target = 7, how many total ways are there to make change? Use the bottom-up DP approach.

hard

A. 6

B. 5

C. 7

D. 4

Solution

1. Step 1: Initialize dp array

   dp[0]=1, dp[1..7]=0 initially.

2. Step 2: Process coin 1

   Update dp for amounts 1 to 7 by adding dp[j-1]: dp=[1,1,1,1,1,1,1,1]

3. Step 3: Process coin 3

   Update dp for j=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

   Update dp for j=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 D

6. Quick Check:

   Count ways with coins {1,3,5} for 7 = 4 [OK]

Hint: Add ways for each coin incrementally [OK]

Common Mistakes:

• Not updating dp correctly for each coin
• Forgetting dp[0] = 1 initialization
• Mixing order of coin processing