<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, amount):
        if amount == 0:
            return 0
    
        memo = {}
        min_coin = 2 ** 31 - 1
        for coin in coins:
            if coin < min_coin:
                min_coin = coin
            memo[coin] = 1
    
        if amount < min_coin:
            return -1
    
        def dp(rem_amount):
            if rem_amount in memo:
                return memo[rem_amount]
            if rem_amount < 0:
                return -1
            if rem_amount < min_coin:
                memo[rem_amount] = -1
                return memo[rem_amount]
    
            min_res = float('inf')
            for i in range(len(coins) - 1, -1, -1):
                coin = coins[i]
                res = dp(rem_amount - coin)
                if res != -1 and res < min_res:
                    min_res = res + 1
    
            memo[rem_amount] = min_res if min_res != float('inf') else -1
            return memo[rem_amount]
    
        memo[amount] = dp( amount)
        return memo[amount]
    
    
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