HMMT 十一月 2010 · 冲刺赛 · 第 19 题

英文解析

19. [ 11 ] How many 8-digit numbers begin with 1, end with 3, and have the property that each successivedigit is either one more or two more than the previous digit, considering 0 to be one more than 9?
    Guts Round
    Answer: 21 Given an 8-digit number a that satifies the conditions in the problem, let a denoteithe difference between its ( i + 1)th and i th digit. Since i ∈ { 1 , 2 } for all 1 ≤ i ≤ 7, we have 7 ≤
    a + a + · · · + a ≤ 14. The difference between the last digit and the first digit of m is 3 − 1 ≡ 2
    1 2 7
    (mod 10), which means a + · · · + a = 12. Thus, exactly five of the a s equal to 2 and the remaining
    1 7 i
    ( )
    two equal to 1. The number of permutations of five 2 s and two 1 s is = 21.7 2