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