21 lines
855 B
Python
21 lines
855 B
Python
def starts_one_ends(n):
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"""
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Given a positive integer n, return the count of the numbers of n-digit
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positive integers that start or end with 1.
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"""
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if n == 1:
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return 1
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# For n-digit numbers:
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# Numbers starting with 1: 1 * 10^(n-1) choices (1 followed by any n-1 digits)
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# Numbers ending with 1: 9 * 10^(n-2) * 1 choices (first digit 1-9, middle n-2 digits 0-9, last digit 1)
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# Numbers starting AND ending with 1: 1 * 10^(n-2) * 1 choices (overlap to subtract)
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# Using inclusion-exclusion:
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# |A or B| = |A| + |B| - |A and B|
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starts_with_1 = 10 ** (n - 1) # 1 followed by any (n-1) digits
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ends_with_1 = 9 * (10 ** (n - 2)) # first digit 1-9, middle (n-2) any, last digit 1
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both = 10 ** (n - 2) # starts with 1 and ends with 1
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return starts_with_1 + ends_with_1 - both |