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

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Coin Change Ways Calculation

What is the output of the following TypeScript code that calculates the total number of ways to make change for amount 5 using coins [1, 2, 3]?

DSA Typescript

function countWays(coins: number[], amount: number): number {
  const dp = Array(amount + 1).fill(0);
  dp[0] = 1;
  for (const coin of coins) {
    for (let i = coin; i <= amount; i++) {
      dp[i] += dp[i - coin];
    }
  }
  return dp[amount];
}

console.log(countWays([1, 2, 3], 5));

A 6

B 5

C 7

D 8

💡 Hint

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

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 total ways problem, why do we initialize dp[0] = 1 before starting the loops?

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] represents the maximum coin value.

D Because dp[0] must be zero to avoid counting invalid ways.

💡 Hint

Think about the base case for making amount zero.

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 TypeScript code produce when run?

DSA Typescript

function countWays(coins: number[], amount: number): number {
  const dp = Array(amount + 1).fill(0);
  dp[0] = 1;
  for (let i = 0; i <= coins.length; i++) {
    for (let j = coins[i]; j <= amount; j++) {
      dp[j] += dp[j - coins[i]];
    }
  }
  return dp[amount];
}

console.log(countWays([1, 2, 3], 5));

A RuntimeError: Negative index access

B SyntaxError: Unexpected token

C Output: 7

D TypeError: Cannot read property 'undefined' of undefined

💡 Hint

Check the loop bounds when accessing coins array.

The loop runs i from 0 to coins.length inclusive, causing coins[i] to be undefined at last iteration.

❓ Predict Output

advanced

Output of Coin Change with Single Coin

What is the output of this code that calculates ways to make amount 10 using only coin 2?

DSA Typescript

function countWays(coins: number[], amount: number): number {
  const dp = Array(amount + 1).fill(0);
  dp[0] = 1;
  for (const coin of coins) {
    for (let i = coin; i <= amount; i++) {
      dp[i] += dp[i - coin];
    }
  }
  return dp[amount];
}

console.log(countWays([2], 10));

A 5

B 1

C 10

D 0

💡 Hint

How many ways to make 10 using only 2s?

Only one combination repeated 5 times: 2+2+2+2+2, so 1 way.

🧠 Conceptual

expert

Why Use Nested Loops in Coin Change DP?

Why does the coin change total ways solution use nested loops: outer loop over coins and inner loop over amounts?

A To ensure each coin is considered in order and combinations are counted without duplicates.

B To count permutations of coins instead of combinations.

C To reduce time complexity from exponential to linear.

D To initialize the dp array with zeros.

💡 Hint

Think about how order of coins affects counting combinations.

Outer loop over coins ensures combinations count without duplicates by processing coins one by one.

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 make a target amount using given coins.

2. Step 2: Differentiate from similar problems

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

3. Final Answer:

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

4. Quick Check:

   Counting ways = total combinations [OK]

Hint: Count combinations, not minimum coins needed [OK]

Common Mistakes:

• Confusing with minimum coin count problem
• Thinking about maximum coin value instead of ways
• Ignoring repeated use of coins

2. Which of the following TypeScript code snippets correctly initializes the DP array for the coin change ways problem for amount 5?

easy

A. const dp = new Array(5).fill(1);

B. const dp = new Array(6).fill(0); dp[0] = 1;

C. const dp = new Array(5).fill(0); dp[1] = 1;

D. const dp = new Array(6).fill(1); dp[0] = 0;

Solution

1. Step 1: Understand DP array size and initialization

   For amount 5, dp array size should be 6 (0 to 5). dp[0] = 1 means one way to make amount 0.

2. Step 2: Check each option

   const dp = new Array(6).fill(0); dp[0] = 1; correctly creates dp of size 6, fills 0, sets dp[0]=1. Others have wrong size or initialization.

3. Final Answer:

   const dp = new Array(6).fill(0); dp[0] = 1; -> Option B

4. Quick Check:

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

Hint: DP array size = amount + 1, dp[0] = 1 [OK]

Common Mistakes:

• Using dp array size equal to amount instead of amount+1
• Not setting dp[0] = 1
• Filling dp with 1 instead of 0 initially

3. What is the output of the following TypeScript code?

const coins = [1, 2];
const amount = 3;
const dp = new Array(amount + 1).fill(0);
dp[0] = 1;
for (const coin of coins) {
  for (let x = coin; x <= amount; x++) {
    dp[x] += dp[x - coin];
  }
}
console.log(dp[amount]);

medium

A. 3

B. 0

C. 1

D. 2

Solution

1. Step 1: Trace dp updates for coin 1

   dp after coin 1: dp[1]=1, dp[2]=1, dp[3]=1 (only using coin 1)

2. Step 2: Trace dp updates for coin 2

   dp[2]+=dp[0]=1+1=2, dp[3]+=dp[1]=1+1=2; final dp = [1,1,2,2]

3. Final Answer:

   2 -> Option D

4. Quick Check:

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

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

Common Mistakes:

• Not updating dp correctly inside nested loops
• Confusing dp[x] with dp[x - coin]
• Missing dp[0] = 1 initialization

4. Find the error in this TypeScript code for coin change ways and select the fix:

const coins = [1, 2, 3];
const amount = 4;
const dp = new Array(amount + 1).fill(0);
dp[0] = 1;
for (let i = 0; i <= coins.length; i++) {
  for (let x = coins[i]; x <= amount; x++) {
    dp[x] += dp[x - coins[i]];
  }
}
console.log(dp[amount]);

medium

A. Use for (const coin of coins) instead of index loop

B. Initialize dp array with 1 instead of 0

C. Change loop to for (let i = 0; i < coins.length; i++)

D. Set dp[amount] = 0 before loops

Solution

1. Step 1: Identify loop index error

   The loop runs i from 0 to coins.length inclusive, causing out-of-bounds access at coins[coins.length].

2. Step 2: Fix loop condition

   Change to i < coins.length to avoid accessing undefined coin.

3. Final Answer:

   Change loop to for (let i = 0; i < coins.length; i++) -> Option C

4. Quick Check:

   Loop index must be less than array length [OK]

Hint: Use < length, not <= length for array loops [OK]

Common Mistakes:

• Using <= in for loop causing out-of-bounds error
• Not initializing dp[0] = 1
• Confusing index and value in loops

5. Given coins = [1, 3, 5] and amount = 7, how many total ways are there to make amount 7 using any number of these coins?

hard

A. 4

B. 8

C. 7

D. 6

Solution

1. Step 1: Initialize dp array and process coin 1

   dp size 8, dp[0]=1. Using coin 1, dp[x]=1 for x=1 to 7.

2. Step 2: Update dp with coin 3

   For x=3 to 7, dp[x]+=dp[x-3]: dp[3]=2, dp[4]=1+dp[1]=2, dp[5]=1+dp[2]=2, dp[6]=1+dp[3]=3, dp[7]=1+dp[4]=3.

3. Step 3: Update dp with coin 5

   For x=5 to 7, dp[5]+=dp[0]=2+1=3, dp[6]+=dp[1]=3+1=4, dp[7]+=dp[2]=3+1=4.

4. Step 4: Final dp array and answer

   Final dp[7] = 4 ways. The combinations are: 7x1, 4x1+1x3, 1x1+2x3, 2x1+1x5.

5. Final Answer:

   4 -> Option A

6. Quick Check:

   Count all combinations with coins [1,3,5] for 7 = 4 [OK]

Hint: Build dp adding ways coin by coin [OK]

Common Mistakes:

• Missing some combinations with multiple coins
• Confusing permutations with combinations
• Incorrect dp updates order