<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:

    def coinChange(self, coins: List[int], amount: int) -> int:

        self.coin_mem = {}

        #coins.sort(reverse=True) # O(nlogn)
        return self.get_coin_change(coins, amount)

    def get_coin_change(self, coins: List[int], amount: int, start: int = 0) -> int:

        key_coin_dct = (start, amount)

        mem_out = self.coin_mem.get(key_coin_dct, None)
        if mem_out is not None:
            return mem_out

        if amount == 0:
            self.coin_mem[key_coin_dct] = 0
            return 0
        elif start == len(coins):
            self.coin_mem[key_coin_dct] = -1
            return -1

        # not use first coin
        out_1 = self.get_coin_change(coins, amount, start=start+1)

        # use first coin
        if amount >= coins[start]:
            out_2 = self.get_coin_change(coins, amount - coins[start], start=start)

            if out_2 == -1:
                self.coin_mem[key_coin_dct] = out_1
                return out_1
            
            out_2 += 1

            if out_1 == -1:
                self.coin_mem[key_coin_dct] = out_2
                return out_2
            else:
                out = min(out_1, out_2)
                self.coin_mem[key_coin_dct] = out
                return out
        
        self.coin_mem[key_coin_dct] = out_1
        return out_1

        """
        # generate all possible permutations: O(n!)
        all_permutations = list(itertools.permutations(coins))
   
        #coins.sort(reverse=False)
        out_coins = None
        for coins_reordered in all_permutations:
            _coins = self.get_coin_change(coins_reordered, amount)
            print(f"{coins_reordered}-{_coins}")
            if out_coins is None:
                out_coins = _coins
            elif _coins != -1 and _coins < out_coins:
                out_coins = _coins
        return out_coins

        """
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