<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 number of combinations that make up that amount</em>. If that amount of money cannot be made up by any combination of the coins, return <code>0</code>.</p>

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

<p>The answer is <strong>guaranteed</strong> to fit into a signed <strong>32-bit</strong> integer.</p>

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

<pre>
<strong>Input:</strong> amount = 5, coins = [1,2,5]
<strong>Output:</strong> 4
<strong>Explanation:</strong> 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
</pre>

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

<pre>
<strong>Input:</strong> amount = 3, coins = [2]
<strong>Output:</strong> 0
<strong>Explanation:</strong> the amount of 3 cannot be made up just with coins of 2.
</pre>

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

<pre>
<strong>Input:</strong> amount = 10, coins = [10]
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>1 &lt;= coins.length &lt;= 300</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 5000</code></li>
	<li>All the values of <code>coins</code> are <strong>unique</strong>.</li>
	<li><code>0 &lt;= amount &lt;= 5000</code></li>
</ul>

class Solution {

public:

    int changeHelper(int amount, const vector<int>& coins, const int &coinCount, map<pair<int, int>, int> &dp_memo, int index)
    {
        if(amount == 0)
            return 1;
        if(amount < 0)
            return 0;

        pair<int, int> d = make_pair(amount, index);

        if(dp_memo.find(d) != dp_memo.end())
            return dp_memo[d];
        int result = 0;
        for(unsigned int i=index;i<coinCount; i++)
        {
            const int &currCoin = coins[i];
            result += changeHelper(amount-currCoin, coins, coinCount, dp_memo, i);
        }
        dp_memo[d] = result;
        return result;
    }
    
    

    int change(int amount, vector<int>& coins) 
    {
        map<pair<int, int>, int> dp_memo;
        sort(coins.rbegin(), coins.rend());
        return changeHelper(amount, coins, coins.size(), dp_memo, 0);
    }
};
24195