<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 recursion(ind, target): 

        #     if ind == 0 :
        #         if target % coins[ind] == 0:
        #             return target // coins[ind] 

                
        #         return float("inf") 

        #     if (ind, target) in dp:
        #         return dp[(ind, target)]
            
        #     notTake = 0 + recursion(ind-1, target)
            
        #     take = float("inf")
        #     if coins[ind] <= target: #not taking amount that is greater than target
        #         #fucntion is counting coins; therefore add 1
        #         take = 1 + recursion(ind, target - coins[ind])
        #     dp[(ind, target)] = min(notTake, take)
        #     return dp[(ind, target)] 
        
        # n = len(coins)
        # dp = {}
        # ans = recursion(n-1, amount)
        # print(1+float("inf"))
        # return ans if ans != float("inf") else -1
        
                
        # TABULATION:

        dp = {}
        for target in range(amount+1):
            if target % coins[0] == 0:
                dp[(0,target)] = target // coins[0]

            else:
                dp[(0,target)] = float("inf")

        #written for ind = 0
        #now write loops in opposite fashion for both of the changing parameters
        n = len(coins)
        for ind in range(1, n):
            for target in range(amount+1):
                notTake = 0 + dp[(ind-1,target)]
            
                take = float("inf")
                if coins[ind] <= target: #not taking amount that is greater than target
                    #fucntion is counting coins; therefore add 1
                    take = 1 + dp[(ind, target - coins[ind])]
                dp[(ind, target)] = min(notTake, take)

        ans = dp[(n-1, amount)]
        return ans if ans != float("inf") else -1
        
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