<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:
    #define DPSolver ios_base::sync_with_stdio(0),cin.tie(0),cout.tie(0);  
    map<int,int> memo;
    bool dpSolverFlag = false;
    int coinChange(vector<int>& coins, int amount, bool notFirstStep = false ) {
        if(!dpSolverFlag){
            DPSolver;
            dpSolverFlag = true; 
        }
        // make it efficient
        auto it = memo.find(amount); 
        if(it != memo.end()) 
            return it -> second; 

        // make it work 
        // base case 
        if(amount == 0)
            return 0; 
        else if (amount < 0){
            memo[amount] = INT_MAX; 
            return INT_MAX; 
        }

        int minCoins = INT_MAX; 
        for(int c : coins)
            minCoins = min(minCoins, coinChange(coins, amount - c, true)); 
        
        if(minCoins == INT_MAX){
            memo[amount] = INT_MAX; 
            if(notFirstStep)
                return INT_MAX; 
            else 
                return -1; 
        }
        else {
            memo[amount] = minCoins + 1; 
            return minCoins + 1; 
        }
    }
};
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