AMC10 2017 B

AMC10 2017 B · Q16

AMC10 2017 B · Q16. It mainly tests Basic counting (rules of product/sum), Base representation.

How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0?

在小于等于2017的正整数的十进制表示中，有多少个包含数字0？

(A) 469 469

(B) 471 471

(D) 478 478

(E) 481 481

Answer

Correct choice: (A)

正确答案：（A）

Solution

It will be easier to count the complementary set. There are 9 one-digit numerals that do not contain the digit 0, 9·9 = 81 two-digit numerals that do not contain the digit 0, 9 · 9 · 9 = 729 three-digit numerals that do not contain the digit 0, and 1 · 9 · 9 · 9 = 729 four-digit numerals starting with 1 that do not contain the digit 0, a total of 1548. All four-digit numerals between 2000 and 2017, inclusive, contain the digit 0. Therefore 2017 − 1548 = 469 numerals in the required range do contain the digit 0.

计算互补集更容易。不含数字0的一位数有9个，不含0的两位数有9·9=81个，不含0的三位数有9·9·9=729个，以1开头的四位数（不含0）有1·9·9·9=729个，总计1548个。2000到2017（包括两端）的所有四位数都包含数字0。因此，2017−1548=469个数字包含数字0。

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