<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 Count(std::vector<int>& coins, int total, int idx, std::vector<std::vector<int>>* const dp) {
        if (total == 0) {
            return 1;
        }
        if (idx == coins.size() || total < 0) {
            return 0;
        }

        if ((*dp)[total][idx] == -1) {
            int cnt = 0;
            for (int i = idx; i < coins.size(); ++i) {
                if (coins[i] > total) {
                    break;
                }

                cnt += Count(coins, total - coins[i], i, dp);
            }

            (*dp)[total][idx] = cnt;
        }

        return (*dp)[total][idx];
    }

    int change(int amount, vector<int>& coins) {
        std::sort(coins.begin(), coins.end());

        std::vector<std::vector<int>> dp(amount + 1, std::vector<int>(coins.size(), -1));
        return Count(coins, amount, 0, &dp);
    }
};
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