<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:


        from functools import cache
        @cache
        def func(s):
            if s ==amount:
                return 0
            if s>amount:
                return float('inf')

            r=float('inf')
            for c in coins:
                r=min(r, 1+func(s+c))
            return r

        r= func(0)    
        return r if r!=float('inf') else -1            






        from functools import cache
        @cache
        def func(idx, s):
            if s==amount:
                return 0
            if s>amount or idx==len(coins):
                return float('inf')
            
            return min(1+func(idx,s+coins[idx]), func(idx+1,s) )
            # res= float('inf')
            # for c in coins:
            #     res=min(res,func(s+c)+1)
            # return res
        res= func(0,0)
        return res if res !=float('inf') else -1

 







        dp=[-1]*(amount+1)
        dp[0]=0

        for c in coins:
            for i in range(amount-c+1):
                if dp[i]!=-1:
                    if dp[i+c]== -1:
                        dp[i+c]=dp[i]+1
                    else:
                        dp[i+c]=min(dp[i+c],dp[i]+1)
        return dp[-1]

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