Happy traveller [Part 1]

There is a play grid NxN; Always square! 

   0  1  2  3
0 [o, o, o, X]
1 [o, o, o, o]
2 [o, o, o, o]
3 [o, o, o, o]


You start from a random point. I mean, you are given the coordinates of your start position in format (row, col). 

And your TASK is to define the number of unique paths to reach position X (always in the top right corner).


From any point you can go only UP or RIGHT.


Implement a function count_paths(N, (row, col)) which returns int; 
Assume input params are always valid.


Example:

count_paths(1, (0, 0))
grid 1x1:

[X]
 You are already in the target point, so return 0


count_paths(2, (1, 0))
grid 2x2:

[o, X]
[@, o]
 You are at point @; you can move UP-RIGHT or RIGHT-UP, and there are 2 possible unique paths here

count_paths(2, (1, 1))
grid 2x2:

[o, X]
[o, @]
 You are at point @; you can move only UP, so there is 1 possible unique path here

count_paths(3, (1, 0))
grid 3x3:

[o, o, X]
[@, o, o]
[o, o, o]
 You are at point @; you can move UP-RIGHT-RIGHT or RIGHT-UP-RIGHT, or RIGHT-RIGHT-UP, and there are 3 possible unique paths here


I think it's pretty clear =)

btw. you can use preloaded Grid class, which constructs 2d array for you. It's very very basic and simple. You can use numpy instead or any other way to produce the correct answer =)
 grid = Grid(2, 2, 0) 
samegrid = Grid.square(2) will give you a grid[2][2], which you can print easily to console. 

print(grid)


[0, 0]
[0, 0]


Enjoy!

You can continue adventures:
Happy traveller [Part 2]

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "import math\n\n# if you look at the matrix of possible paths...\n# [ 1,  1,  1,  0]\n# [ 4,  3,  2,  1]\n# [10,  6,  3,  1]\n# [20, 10,  4,  1]\n# ...the diagonals reflect Pascal's triangle.\n#\n# For example, 4C1 = 4, 4C2 = 6, 4C3 = 4\n# Each cell (row, col) has possible paths = dC(l-row) where:\n#    d = the ordinal of the diagonal (zero-indexed)\n#    l = the length of the diagonal\n#\n# We can calculate d and l like so:\n#    d = row + n - 1 - col\n#    l = d + 1\n#\n# Combined, we get:\n#    (row + n - 1 - col) C (n - 1 - col)\ndef count_paths(N, coords):\n    row, col = coords\n    if row == 0 and col == N-1:\n        return 0\n    \n    inverse_col = N - 1 - col\n    return choose(row + inverse_col, inverse_col)\n\n# aCb = a!/(a-b)!b!\ndef choose(n,k):\n    if k > n:\n        return 0\n    if k > n-k:\n        k = n-k\n    product = 1\n    k_list = list(range(2, k+1))\n    for i in range(n, n-k, -1):\n        if len(k_list) > 0:\n            k_index = len(k_list)-1\n            k = k_list[k_index]\n            if i % k == 0:\n                i = i // k\n                del k_list[k_index]\n        product *= i\n    for k in k_list:\n        product //= k\n    return product"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'count_paths'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 5360