Coin Change Total Number of Ways in DSA Rust

Mental Model

We want to find how many different ways we can make a total amount using given coin values, counting all combinations.

Analogy: Imagine you have different types of building blocks (coins) and you want to build a tower of a certain height (amount). You count all the ways to stack blocks to reach that height, using any number of each block.

coins: [1, 2, 3]
amount: 4

ways array (index = amount):
[1, 0, 0, 0, 0]
 ↑
Initially, 1 way to make 0 amount (use no coins).

Dry Run Walkthrough

Input: coins: [1, 2, 3], amount: 4

Goal: Find total number of ways to make amount 4 using coins 1, 2, and 3.

Step 1: Start with ways array: ways[0] = 1 (one way to make 0), others 0

ways: [1, 0, 0, 0, 0]

Why: We know there is exactly one way to make amount 0: use no coins.

Step 2: Use coin 1: update ways for amounts 1 to 4 by adding ways[amount - 1]

ways: [1, 1, 1, 1, 1]

Why: With coin 1, each amount can be made by adding coin 1 to ways of amount-1.

Step 3: Use coin 2: update ways for amounts 2 to 4 by adding ways[amount - 2]

ways: [1, 1, 2, 2, 3]

Why: Coin 2 adds new combinations for amounts 2, 3, 4 based on ways to make amount-2.

Step 4: Use coin 3: update ways for amounts 3 to 4 by adding ways[amount - 3]

ways: [1, 1, 2, 3, 4]

Why: Coin 3 adds new combinations for amounts 3 and 4 based on ways to make amount-3.

Result:

ways: [1, 1, 2, 3, 4]
Total ways to make 4 = 4

Annotated Code

DSA Rust

fn coin_change_ways(coins: &[usize], amount: usize) -> usize {
    let mut ways = vec![0; amount + 1];
    ways[0] = 1; // one way to make amount 0

    for &coin in coins {
        for current_amount in coin..=amount {
            ways[current_amount] += ways[current_amount - coin];
        }
    }

    ways[amount]
}

fn main() {
    let coins = vec![1, 2, 3];
    let amount = 4;
    let total_ways = coin_change_ways(&coins, amount);
    println!("Total ways to make {} with coins {:?} = {}", amount, coins, total_ways);
}

ways[0] = 1; // one way to make amount 0

Initialize ways array with base case: one way to make zero amount

for &coin in coins {

Iterate over each coin to build up ways array

for current_amount in coin..=amount {

For each amount from coin value up to target amount

ways[current_amount] += ways[current_amount - coin];

Add ways to make (current_amount - coin) to ways[current_amount] to count new combinations

Output

Total ways to make 4 with coins [1, 2, 3] = 4

Complexity Analysis

Time: O(n * amount) because we iterate over each coin and for each coin over all amounts up to target

Space: O(amount) because we store ways counts for all amounts from 0 to target

vs Alternative: Compared to recursive brute force exponential approach, this dynamic programming approach is much faster and uses less repeated work

Edge Cases

amount = 0

Returns 1 because there is exactly one way to make zero amount (use no coins)

DSA Rust

ways[0] = 1; // one way to make amount 0

coins array empty

Returns 0 for any positive amount because no coins to make change

DSA Rust

for &coin in coins { ... } // loop skipped if coins empty

coins contain duplicates

Duplicates do not affect correctness but may cause counting same combinations multiple times if not handled; here duplicates are processed as is

DSA Rust

for &coin in coins { ... } // processes each coin independently

When to Use This Pattern

When asked to count the number of ways to make a total using given coin denominations, use the coin change total number of ways pattern with dynamic programming to efficiently count combinations.

Common Mistakes

Mistake: Using a recursive approach without memoization causing exponential time
Fix: Use bottom-up dynamic programming with a ways array to store intermediate results

Mistake: Updating ways array in wrong order causing overcounting
Fix: Iterate coins outer loop and amounts inner loop from coin to amount to avoid counting permutations

Summary

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

Use when you need to find how many combinations of coins sum to a value.

The key insight is to build up the count for each amount by adding ways from smaller amounts using each coin.

Practice

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

easy

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

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

C. The maximum value of coins that can be used without exceeding 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 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 B

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 Rust code snippets correctly initializes a vector to store ways to make amounts up to `target` with 0 ways initially except for 0 amount which has 1 way?

easy

A. let mut ways = vec![0; target + 1]; ways[0] = 1;

B. let mut ways = vec![1; target]; ways[0] = 0;

C. let mut ways = vec![0; target]; ways[0] = 1;

D. let mut ways = vec![1; target + 1]; ways[0] = 0;

Solution

1. Step 1: Initialize vector size correctly

   Vector size must be target + 1 to include amount 0 to target.

2. Step 2: Set base case for amount 0

   ways[0] = 1 means one way to make amount 0 (using no coins).

3. Final Answer:

   let mut ways = vec![0; target + 1]; ways[0] = 1; -> Option A

4. Quick Check:

   Vector size = target+1, ways[0] = 1 [OK]

Hint: ways vector size = target+1, ways[0] = 1 [OK]

Common Mistakes:

• Using vector size target instead of target+1
• Setting ways[0] to 0 instead of 1
• Initializing all elements to 1 incorrectly

3. What is the output of the following Rust code snippet for coins = [1, 2] and target = 3?

let coins = vec![1usize, 2usize];
let target: usize = 3;
let mut ways = vec![0; target + 1];
ways[0] = 1;
for &coin in &coins {
    for amount in coin..=target {
        ways[amount] += ways[amount - coin];
    }
}
println!("{}", ways[target]);

medium

A. 4

B. 1

C. 2

D. 3

Solution

1. Step 1: Calculate ways using coin 1

   ways after coin 1: ways[0]=1, ways[1]=1, ways[2]=1, ways[3]=1

2. Step 2: Update ways using coin 2

   ways[2] += ways[0] => 1+1=2; ways[3] += ways[1] => 1+1=2; final ways = [1,1,2,2]

3. Final Answer:

   2 -> Option C

4. Quick Check:

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

Hint: Add ways for each coin incrementally [OK]

Common Mistakes:

• Not updating ways correctly for each coin
• Confusing ways count with minimum coins
• Off-by-one errors in loops

4. Identify the bug in this Rust code snippet for coin change ways calculation:

let coins = vec![1usize, 2usize, 3usize];
let target: usize = 4;
let mut ways = vec![0; target + 1];
ways[0] = 1;
for &coin in &coins {
    for amount in 1usize..=target {
        if amount >= coin {
            ways[amount] += ways[amount - coin];
        }
    }
}
println!("{}", ways[target]);

medium

A. The inner loop should start from coin, not 1.

B. ways[0] should be initialized to 0.

C. coins vector should be mutable.

D. The outer loop should iterate over amounts, not coins.

Solution

1. Step 1: Check inner loop range

   Starting inner loop from 1 causes unnecessary checks; it should start from coin value to avoid redundant iterations.

2. Step 2: Confirm initialization and loops

   ways[0] = 1 is correct; coins vector does not need to be mutable; outer loop over coins is correct.

3. Final Answer:

   The inner loop should start from coin, not 1. -> Option A

4. Quick Check:

   Inner loop start = coin [OK]

Hint: Inner loop starts at coin value to optimize [OK]

Common Mistakes:

• Starting inner loop from 1 instead of coin
• Incorrect initialization of ways[0]
• Misunderstanding loop roles

5. Given coins = [1, 3, 5] and target = 7, how many total ways are there to make the target amount? Use the coin change total number of ways approach.

hard

A. 6

B. 5

C. 7

D. 4

Solution

1. Step 1: Initialize ways vector

   ways = [1,0,0,0,0,0,0,0] for amounts 0 to 7.

2. Step 2: Update ways with coin 1

   ways after coin 1: ways[i] = 1 for all i from 0 to 7 (only using 1s).

3. Step 3: Update ways with coin 3

   ways[3]+=ways[0]=1, ways[4]+=ways[1]=1, ways[5]+=ways[2]=1, ways[6]+=ways[3]=2, ways[7]+=ways[4]=2; ways now: [1,1,1,2,2,2,3,3]

4. Step 4: Update ways with coin 5

   ways[5] += ways[0] => 2 + 1 = 3; ways[6] += ways[1] => 3 + 1 = 4; ways[7] += ways[2] => 3 + 1 = 4; final ways: [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 incrementally for each coin [OK]

Common Mistakes:

• Not updating ways correctly for each coin
• Forgetting to include ways[0] = 1
• Mixing up order of loops