Coin Change II

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, $10$ and $20$ cents, and you want to represent the total amount of $30$ cents using these coins. There are only two ways to do this, you can use either of the following combinations:

• 3 coins of $10$ cents: $10+10+10=30$.

• 1 coin of $10$ cents and 1 coin of $20$ cents: $10+20=30.$

Constraints:

• 1 <= coins.length <= 300

• 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:

Try it yourself

Implement your solution in the following playground.

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, $10+20$ cents add up to $30$ cents, and so do $20+10$. 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 $n$ coins, the second time, we can use $n−1$ coins, that is, by excluding the largest one, and so on. This way, it is not ...