# 518 Coin Change 2

## 518. [Coin Change 2](https://leetcode.com/problems/coin-change-2/description/)

## 1. Question

You are given coins of different denominations and a total amount of money. Write a function to compute the number of combinations that make up that amount. You may assume that you have infinite number of each kind of coin.

**Note:**You can assume that

* 0 <= amount <= 5000
* 1 <= coin <= 5000
* the number of coins is less than 500
* the answer is guaranteed to fit into 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
```

## 2. Implementation

**(1) 完全背包**

思路: 注意这里要求的是**组合而不是排列**, 所以第一层的for loop是coins数组中每个可能的coin，而第二层for loop是amount。如果要求排列的话，两个for loop的顺序要反过来。dp\[i]表示对于amount为i，需要的coins的combination

```java
class Solution {
    public int change(int amount, int[] coins) {
        int[] dp = new int[amount + 1];
        dp[0] = 1;
        for (int j = 0; j < coins.length; j++) {
            for (int i = 1; i <= amount; i++) {
                if (coins[j] <= i) {
                    dp[i] += dp[i - coins[j]];
                }
            }
        }
        return dp[amount];
    }
}
```

## 3. Time & Space Complexity

**完全背包:** 时间复杂度O(n \* amount), 空间复杂度O(amount)