# 0518. Coin Change 2

<https://leetcode.com/problems/coin-change-2>

## Description

You are given an integer array `coins` representing coins of different denominations and an integer `amount` representing a total amount of money.

Return *the number of combinations that make up that amount*. If that amount of money cannot be made up by any combination of the coins, return `0`.

You may assume that you have an infinite number of each kind of coin.

The answer is **guaranteed** to fit into a signed **32-bit** integer.

**Example 1:**

```
**Input:** amount = 5, coins = [1,2,5]
**Output:** 4
**Explanation:** there are four ways to make up the amount:
5=5
5=2+2+1
5=2+1+1+1
5=1+1+1+1+1
```

**Example 2:**

```
**Input:** amount = 3, coins = [2]
**Output:** 0
**Explanation:** the amount of 3 cannot be made up just with coins of 2.
```

**Example 3:**

```
**Input:** amount = 10, coins = [10]
**Output:** 1
```

**Constraints:**

* `1 <= coins.length <= 300`
* `1 <= coins[i] <= 5000`
* All the values of `coins` are **unique**.
* `0 <= amount <= 5000`

## ac

This is a classic knapsack problem.

dp\[i]\[j] : the number of combinations to make up amount j by using the first i types of coins

State transition:

* not using the ith coin, only using the first i-1 coins to make up amount j, then we have dp\[i-1]\[j] ways.
* using the ith coin, since we can use unlimited same coin, we need to know how many ways to make up amount j - coins\[i-1] by using first i coins(including ith), which is dp\[i]\[j-coins\[i-1]]

Initialization: dp\[i]\[0] = 1

```java
public int change(int amount, int[] coins) {
        int[][] dp = new int[coins.length+1][amount+1];
        dp[0][0] = 1;

        for (int i = 1; i <= coins.length; i++) {
            dp[i][0] = 1;
            for (int j = 1; j <= amount; j++) {
                dp[i][j] = dp[i-1][j] + (j >= coins[i-1] ? dp[i][j-coins[i-1]] : 0);
            }
        }
        return dp[coins.length][amount];
    }

// Now we can see that dp[i][j] only rely on dp[i-1][j] and dp[i][j-coins[i]], then we can optimize the space by only using one-dimension array.
public int change(int amount, int[] coins) {
        int[] dp = new int[amount + 1];
        dp[0] = 1;
        for (int coin : coins) {
            for (int i = coin; i <= amount; i++) {
                dp[i] += dp[i-coin];
            }
        }
        return dp[amount];
    }
```

\[Why an inner loop for coins doensn't work?]\(<https://leetcode.com/problems/coin-change-2/discuss/176706/Beginner-Mistake:-Why-an-inner-loop-for-coins-doensn't-work-Java-Soln/306232>)

> To get the correct answer, the correct dp definition should be dp\[i]\[j]="number of ways to get sum 'j' using 'first i' coins". Now when we try to traverse the 2-d array row-wise by keeping only previous row array(to reduce space complexity), we preserve the above dp definition as dp\[j]="number of ways to get sum 'j' using 'previous /first i coins' " but when we try to traverse the 2-d array column-wise by keeping only the previous column, the meaning of dp array changes to dp\[j]="number of ways to get sum 'j' using 'all' coins".


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