number-of-ways-to-reach-a-position-after-exactly-k-steps.py

# Time:  O(k)
# Space: O(k)

# combinatorics
class Solution(object):
    def numberOfWays(self, startPos, endPos, k):
        """
        :type startPos: int
        :type endPos: int
        :type k: int
        :rtype: int
        """
        MOD = 10**9+7
        fact, inv, inv_fact = [[1]*2 for _ in xrange(3)]
        def nCr(n, k):
            while len(inv) <= n:  # lazy initialization
                fact.append(fact[-1]*len(inv) % MOD)
                inv.append(inv[MOD%len(inv)]*(MOD-MOD//len(inv)) % MOD)  # https://cp-algorithms.com/algebra/module-inverse.html
                inv_fact.append(inv_fact[-1]*inv[-1] % MOD)
            return (fact[n]*inv_fact[n-k] % MOD) * inv_fact[k] % MOD

        r = k-abs(endPos-startPos)
        return nCr(k, r//2) if r >= 0 and r%2 == 0 else 0