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coin-change-ii.cpp

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+// Author: Metin Furkan Amarat
+// Question Link: https://leetcode.com/problems/coin-change-ii/
+// Time Complexity and Space Complexity: O(N * amount)
+
+class Solution {
+
+public:
+    int change(int amount, vector<int>& coins) {
+        int n = coins.size();
+        // Initialise the dp array, this is padded on both dimensions, all values initialised as -1, 
+        // so we can differentiate between true 0 values and cells which have not yet been calculated.
+        vector<vector<int>> dp(n + 1, vector<int>(amount + 1, -1)); 
+        return calculate(0, amount, coins, dp);
+    }
+    int calculate(int i, int x, vector<int>& coins, vector<vector<int>> &dp ){
+        // i --> 
+        // The function is recursive, and calls itself with the amount - coins[i].
+        // Therefore, if we hit 0, we have found a legitimate way of reaching the amount using the coins we have taken so far.
+        if (x == 0)
+            return 1;
+        // Ran out of coins to pick, still have an amount. No new ways.
+        if (i == coins.size())
+            return 0;
+        // Lastly, if we are in a situation we calculated before, we just return the value.
+        if(dp[i][x] != -1)
+            return dp[i][x];
+
+        int take = 0; // The number of combinations we have if we take coins[i].
+        if(x >= coins[i]) // We take the coin only if we have an amount large enough.
+        {
+            take = calculate(i, x - coins[i], coins, dp);
+        }
+        int skip = calculate(i + 1, x, coins, dp); // Number of combinations we have if we do not take coins[i], moving on with coins[i + 1].
+        return dp[i][x] = take + skip; // assign the value to the 2d array cell.
+    }
+};