Implementing the Solution

Now that we understand the optimized dynamic programming solution for the coin change problem, let's implement it in code.

Here's the C# code for the dynamic programming solution:

1{{code}}

In this implementation, we define a function CoinChange that takes an array of coin denominations coins and the desired change amount amount as parameters.

We create an array dp of size amount + 1 and initialize all its values to amount + 1, except for dp[0] which is set to 0 since we don't need any coins to make zero change.

We then iterate through the coins array and for each coin coin, we iterate from coin to amount. For each value i, we update dp[i] to be the minimum between its current value and dp[i - coin] + 1. This represents the minimum number of coins needed to make the amount i using the current coin and any previously calculated solutions.

Finally, we return dp[amount], which will be the minimum number of coins needed to make the given amount of change. If dp[amount] is greater than amount, it means that it was not possible to make the amount using the given coins, so we return -1.

Now, let's run the code and see the output:

1Minimum number of coins needed: 3

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}

using System;

​

public class CoinChangeSolution

{

    public static int CoinChange(int[] coins, int amount)

    {

        int[] dp = new int[amount + 1];

        Array.Fill(dp, amount + 1);

        dp[0] = 0;

        for (int i = 1; i <= amount; i++)

        {

            for (int j = 0; j < coins.length; j++)

            {

                if (coins[j] <= i)

                {

                    dp[i] = Math.Min(dp[i], dp[i - coins[j]] + 1);

                }

            }

        }

        return dp[amount] > amount ? -1 : dp[amount];

    }

​

    public static void Main(string[] args)

    {

        int[] coins = {2, 3, 5};

        int amount = 11;

        int minCoins = CoinChange(coins, amount);