<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:
        table = {}

        def dfs(amount):
            if amount<0:
                return float("inf")
            if amount == 0:
                return 0
            
            if amount in table:
                return table[amount]

            table[amount] = min(dfs(amount-coin)+1 for coin in coins)
            return table[amount]

        result = dfs(amount)

        return -1 if result==float("inf") else result
        












        table = {}
        def dp(amount):
            if amount<0:
                return float("inf")
            if amount==0:
                return 0
            
            if amount in table:
                return table[amount]
            
            table[amount] = min(dp(amount-coin)+1 for coin in coins)
            return table[amount]
        
        result = dp(amount)
        return -1 if result == float("inf") else result









        
        table = {} 

        def dp(amount):
            if amount < 0:
                return float("inf")

            if amount==0:
                return 0

            if amount in table:
                return table[amount]

            table[amount] = min(dp(amount-coin)+1 for coin in coins)

            return table[amount]

        result = dp(amount)

        return -1 if result == float("inf") else result 

        # table = {} 

        # def dp(amount):
        #     if amount == 0: 
        #         return 0 
            
        #     if amount <0 :
        #         return float("inf")

        #     if amount in table:
        #         return table[amount]
            
        #     table[amount] = min(dp(amount-coin)+1 for coin in coins)

        #     return table[amount]
        
        # result = dp(amount)

        # return -1 if result == float("inf") else result

        # table = {}

        # def dp(amount):
        #     if amount < 0:
        #         return float("inf")
            
        #     if amount == 0:
        #         return 0
            
        #     if amount in table: 
        #         return table[amount]

        #     table[amount] = min(dp(amount-coin)+1 for coin in coins)

        #     return table[amount]
        
        # result = dp(amount)

        # return -1 if result == float("inf") else result
31596