Implement Parentheses Generator (#5096)

* chore: add `ParenthesesGenerator` to `DIRECTORY.md`

* feat: implement Parentheses Generator

* ref: change `ParenthesesGenerator`s method to `static`

* ref: use parametrized tests

* ref: handling exception when `n < 0`

* chore: update docstrings

* ref: make `ParenthesesGenerator` to be a proper utility

* chore(docs): add private constructor docstring

* ref(tests): move bad name suggestions

* style: remove reduntant comments

---------

Co-authored-by: Piotr Idzik <65706193+vil02@users.noreply.github.com>

src/main/java/com/thealgorithms/backtracking/ParenthesesGenerator.java

@@ -0,0 +1,50 @@
+package com.thealgorithms.backtracking;
+
+import java.util.ArrayList;
+import java.util.List;
+
+/**
+ * This class generates all valid combinations of parentheses for a given number of pairs using backtracking.
+ */
+public final class ParenthesesGenerator {
+    private ParenthesesGenerator() {
+    }
+
+    /**
+     * Generates all valid combinations of parentheses for a given number of pairs.
+     *
+     * @param n The number of pairs of parentheses.
+     * @return A list of strings representing valid combinations of parentheses.
+     * @throws IllegalArgumentException if n is less than 0.
+     */
+    public static List<String> generateParentheses(final int n) {
+        if (n < 0) {
+            throw new IllegalArgumentException("The number of pairs of parentheses cannot be nagative");
+        }
+        List<String> result = new ArrayList<>();
+        generateParenthesesHelper(result, "", 0, 0, n);
+        return result;
+    }
+
+    /**
+     * Helper function for generating all valid combinations of parentheses recursively.
+     *
+     * @param result  The list to store valid combinations.
+     * @param current The current combination being formed.
+     * @param open    The number of open parentheses.
+     * @param close   The number of closed parentheses.
+     * @param n       The total number of pairs of parentheses.
+     */
+    private static void generateParenthesesHelper(List<String> result, final String current, final int open, final int close, final int n) {
+        if (current.length() == n * 2) {
+            result.add(current);
+            return;
+        }
+        if (open < n) {
+            generateParenthesesHelper(result, current + "(", open + 1, close, n);
+        }
+        if (close < open) {
+            generateParenthesesHelper(result, current + ")", open, close + 1, n);
+        }
+    }
+}