CoinChange2.py

class Solution(object):
    def change(self, amount, coins):
        """
        :type amount: int
        :type coins: List[int]
        :rtype: int
        """
        if amount == 0:
            return 1

        #Let us create a DP array. 
        dp = []
        for i in range(len(coins) + 1):
            dp.append([0] * (amount+1)) #The reason why we intialize with 0 is worst case, we wont be able to
                                # make the amount.
            dp[i][0] = 1

        for i in range(1,len(dp)):
            for j in range(1, len(dp[0])):
                # If the amount is less that coin being considered, we will skip 
                # choosing the coin. This is by basically using the value exactly
                # above as it is the value that we have in hand for generating the 
                # same amount without using the current coin.
                if j < coins[i-1]:
                    dp[i][j] = dp[i-1][j]
                    
                else:
                    #Given that we are calculating the total amount, we can simply
                    #calculate the total amount generated by using the coin and not
                    #using it.
                    dp[i][j] = dp[i-1][j] + dp[i][j-coins[i-1]] #dp[i][j-coins[i-1]] <-- remaining amount using the same coin.
        #print(dp)
        return dp[-1][-1]