""" A dominating number is a positive integer that has more than half of its digits equal. For example, 2022is a dominating number because three of its four digits are equal to 2. But 2021 is not a dominating number. Let D(N) be how many dominating numbers are less than 10^N . For example, D(4)=603 and D(10)=21893256 . Find D(2022). Give your answer modulo 1_000_000_007. """ import time def timer(func): def wrapper(*args, **kwargs): start_time = time.time() result = func(*args, **kwargs) end_time = time.time() elapsed_time = end_time - start_time print(f"{func.__name__} time: {elapsed_time:.6f} seconds") return result return wrapper @timer def solve(N: int = 2022, MOD: int = 10**9 + 7): # 预处理阶乘和逆阶乘到 N+1 fact = [1] * (N + 2) for i in range(1, N + 2): fact[i] = fact[i - 1] * i % MOD inv_fact = [1] * (N + 2) inv_fact[N + 1] = pow(fact[N + 1], MOD - 2, MOD) for i in range(N, -1, -1): inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD # 组合数函数 def C(n: int, k: int) -> int: if k < 0 or k > n: return 0 return fact[n] * inv_fact[k] % MOD * inv_fact[n - k] % MOD # 预处理 9 的幂(到 N+2 足够) pow9 = [1] * (N + 3) for i in range(1, N + 3): pow9[i] = pow9[i - 1] * 9 % MOD ans = 0 K_max = (N - 1) // 2 for k in range(0, K_max + 1): term = pow9[k + 1] * (C(N + 1, k + 1) - C(2 * k + 1, k + 1)) % MOD ans = (ans + term) % MOD return ans def main(): n = int(input("N:")) mod = int(input("mod:") or (10**9 + 7)) print(solve(n, mod)) if __name__ == "__main__": main()