<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:
    vector<vector<int>>dp;
    int solve(int amount, vector<int>&coins, int index) {
        if (amount == 0) return 1;
        if (amount < 0) return 0;
        if (index >= coins.size()) return 0;
        int res = 0;
        if (dp[amount][index] != -1) return dp[amount][index];
        int n_amount = amount;
        while(n_amount >= 0) {
            res += solve(n_amount, coins, index+1);
            n_amount -= coins[index];
        }
        dp[amount][index] = res;
        return res;
    }

    int change(int amount, vector<int>& coins) {
        dp = vector<vector<int>>(amount+5, vector<int>(coins.size()+2, -1));
        sort(coins.begin(), coins.end());
        int res = solve(amount, coins, 0);
        return res;
    }
};
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