Add euler project problem 15 additional solution (#12774)

* Add euler project problem 15 additional solution by explicitly counting the paths.

* Update sol2.py

* updating DIRECTORY.md

* updating DIRECTORY.md

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---------

Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>
Co-authored-by: MaximSmolskiy <MaximSmolskiy@users.noreply.github.com>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: cclauss <3709715+cclauss@users.noreply.github.com>

Author: 4 people

DIRECTORY.md

@@ -986,6 +986,7 @@
     * [Sol2](project_euler/problem_014/sol2.py)
   * Problem 015
     * [Sol1](project_euler/problem_015/sol1.py)
+    * [Sol2](project_euler/problem_015/sol2.py)
   * Problem 016
     * [Sol1](project_euler/problem_016/sol1.py)
     * [Sol2](project_euler/problem_016/sol2.py)

project_euler/problem_015/sol2.py

@@ -0,0 +1,32 @@
+"""
+Problem 15: https://projecteuler.net/problem=15
+
+Starting in the top left corner of a 2x2 grid, and only being able to move to
+the right and down, there are exactly 6 routes to the bottom right corner.
+How many such routes are there through a 20x20 grid?
+"""
+
+
+def solution(n: int = 20) -> int:
+    """
+    Solve by explicitly counting the paths with dynamic programming.
+
+    >>> solution(6)
+    924
+    >>> solution(2)
+    6
+    >>> solution(1)
+    2
+    """
+
+    counts = [[1 for _ in range(n + 1)] for _ in range(n + 1)]
+
+    for i in range(1, n + 1):
+        for j in range(1, n + 1):
+            counts[i][j] = counts[i - 1][j] + counts[i][j - 1]
+
+    return counts[n][n]
+
+
+if __name__ == "__main__":
+    print(solution())