<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>amount</code> representing a total amount of money.</p>

<p>Return <em>the fewest number of coins that you need to make up that amount</em>. If that amount of money cannot be made up by any combination of the coins, return <code>-1</code>.</p>

<p>You may assume that you have an infinite number of each kind of coin.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> coins = [1,2,5], amount = 11
<strong>Output:</strong> 3
<strong>Explanation:</strong> 11 = 5 + 5 + 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> coins = [2], amount = 3
<strong>Output:</strong> -1
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> coins = [1], amount = 0
<strong>Output:</strong> 0
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= coins.length &lt;= 12</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>0 &lt;= amount &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution {
public:
    vector<vector<int>> memo;
    int getChanges(vector<int> &nums, int target, int idx) {
        if (target == 0) {
            return 0;
        }
        if (memo[idx][target] != -1) {
            return memo[idx][target];
        }
        if (idx >= nums.size() || target < 0) {
            return INT_MAX;
        }
        // take it 
        int count1 = INT_MAX, count2 = INT_MAX;
        if (nums[idx] <= target) {
            int res = getChanges(nums, target - nums[idx], idx);
            if (res != INT_MAX) {
                count1 = 1 + res;
            }
        }
        
        count2 = getChanges(nums, target, idx + 1);
        return memo[idx][target] = min({ count1, count2 });
    }
    
    int coinChange(vector<int>& coins, int amount) {
        int l = coins.size();
        memo.assign(l+1, vector<int>(amount+1, -1));
        int ans = getChanges(coins, amount, 0);
        return ans == INT_MAX ? -1 : ans;
    }
};
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