<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:
        def collect(idx: int, remaining: int) -> int:
            if idx == 0:
                if remaining % coins[idx] == 0:
                    return remaining // coins[idx]
                
                return float("inf")
            
            if dp[idx][remaining] != -1:
                return dp[idx][remaining]
            
            pick = float("inf")
            notPick = collect(idx-1, remaining)
            if coins[idx] <= remaining:
                pick = 1 + collect(idx, remaining-coins[idx])
            
            dp[idx][remaining] = min(notPick, pick)
            return dp[idx][remaining]
        
        N = len(coins)
        dp = [[-1]*(amount+1) for _ in range(N)]
        result = collect(N-1, amount)
        
        return result if result != float("inf") else -1

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