from itertools import combinations
blocked = {(1, 1), (2, 3)}
count = 0
for right_positions in combinations(range(8), 4):
    right_positions = set(right_positions)
    x = y = 0
    valid = True
    for step in range(8):
        if step in right_positions:
            x += 1
        else:
            y += 1
        if (x, y) in blocked:
            valid = False
            break
    if valid:
        assert (x, y) == (4, 4)
        count += 1
result = {"valid_paths": count}
import json
print(json.dumps(result, ensure_ascii=False, sort_keys=True))