<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 change(int amount, const vector<int>& coins) {
		if ((amount == 0) && (!coins.empty()))
			return 1;

		auto sortedCoins = coins;
		std::sort(sortedCoins.begin(), sortedCoins.end(), std::greater<int>());
		return changeImpl(amount, sortedCoins, 0);
	}

	int changeImpl(int amount, const vector<int>& coins, int coinIndex) {
		auto itSolution = _solutions.find(amount);
		if (itSolution != _solutions.end())
		{
			auto itSolutionEx = itSolution->second.find(coinIndex);
			if (itSolutionEx != itSolution->second.end())
				return itSolutionEx->second;
		}
		
		if (coinIndex == coins.size() - 1)
		{
			int result = (amount % coins[coins.size() - 1] == 0) ? 1 : 0;
			_solutions[amount][coinIndex] = result;
			return result;
		}

		int result = 0;
		if (amount % coins[coinIndex] == 0)
			result += 1;

		for (auto temp = 0; temp < amount; temp += coins[coinIndex])
		{
			result += changeImpl(amount - temp, coins, coinIndex + 1);
		}
		_solutions[amount][coinIndex] = result;
		return result;
	}
private:
	std::map<int, std::map<int, int>> _solutions;
};
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