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Grokking Dynamic Programming Interview

Coin Change II

Explore how to count the number of combinations to make a given amount using unique coins with unlimited supply. Learn to implement naive recursive and optimized dynamic programming solutions, including top-down memoization and bottom-up tabulation methods. Understand time and space complexity improvements and how to apply the unbounded knapsack pattern for efficient problem-solving.

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Statement

Suppose you are given a list of coins and a certain amount of money. Each coin in the list is unique and of a different denominationA denomination is a unit of classification for the stated or face value of financial instruments such as currency notes or coins.. You are required to count the number of ways the provided coins can sum up to represent the given amount. If there is no possible way to represent that amount, then return 0.

Note: You may assume that for each combination you make, you have an infinite number of each coin. In simpler terms, you can use a specific coin as many times as you want.

Let's say you have only two coins, 1010 and 2020 cents, and you want to represent the total amount of 3030 cents using these coins. There are only two ways to do this, you can use either of the following combinations:

  • 3 coins of 1010 cents: 10+10+10=3010+10+10=30.

  • 1 coin of 1010 cents and 1 coin of 2020 cents: 10+20=30.10+20=30.

Constraints:

  • 1 <= coins.length <= 70

  • 1 <= coins[i] <= 5000

  • All the coins have a unique value.

  • 0 <= amount <= 5000

Examples

Let's see a few more examples to get a better understanding of the problem statement:

No.

Coins

Amount

Number of ways

1

[1, 2, 5]

7

6

2

[1, 5, 10, 25]

10

4

3

[7]

9

0

4

[9,10,11]

0

1

Try it yourself

Implement your solution in the following playground.

Java
usercode > CountWays.java
// Importing required functions
import java.util.*;
import java.util.stream.*;
public class CountWays{
    public static long countWays (int[] coins, int amount) {
        // Replace this placeholder return statement with your code
        return -1;
    }
}
Coin Change II

Note: If you clicked the “Submit” button and the code timed out, this means that your solution needs to be optimized in terms of running time.

Hint: Use dynamic programming and see the magic.

Solution

We will first explore the naive recursive solution to this problem and then see how it can be improved using the Unbounded Knapsack dynamic programming pattern.

Naive approach

A naive approach to solve this problem would be to make all the possible combinations of coins or generate all subsets of coin denominations that sum up to the required amount.

While making the combinations, a point to keep in mind is that we should try to avoid repeatedly counting the same combinations. For example, 10+2010+20 cents add up to 3030 cents, and so do 20+1020+10. In the context of combinations, both these sequences are the same, and we will count them as one combination. We can achieve this by using a subset of coins every time we consider them for combinations. The first time, we can use all nn coins, the second time, we can use n−1n−1 ...