AMC8 2016

AMC8 2016 · Q17

AMC8 2016 · Q17. It mainly tests Basic counting (rules of product/sum), Inclusion–exclusion (basic).

An ATM password at Fred's Bank is composed of four digits from 0 to 9, with repeated digits allowable. If no password may begin with the sequence 9, 1, 1, then how many passwords are possible?

Fred 银行的 ATM 密码由 0 到 9 的四位数字组成，允许重复数字。如果密码不得以序列 9,1,1 开头，则可能有多少个密码？

(A) 30 30

(B) 7290 7290

(D) 9990 9990

(E) 9999 9999

Answer

Correct choice: (D)

正确答案：（D）

Solution

If there were no restrictions, then $10^4$ passwords would be possible. Among these, 10 passwords begin with 9 1 1, and have 10 options for the fourth digit. Thus $10^4 - 10 = 9990$ passwords satisfy the condition.

若没有任何限制，则可能的密码有 $10^4$ 个。在这些密码中，有 10 个以 9 1 1 开头，并且第四位有 10 种选择。因此满足条件的密码共有 $10^4 - 10 = 9990$ 个。

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