Question

从 $1$ 到 $1000$ 之间，有多少个整数不含数字 $1$ ？

How many whole numbers between $1$ and $1000$ do not contain the digit $1?$

$512$

$648$

$720$

$728$

$800$

按位数分类计数，或者把数补前导零

Count by number of digits, or pad numbers with leading zeroes.

三个补零后的数位各有 $9$ 种选择（禁止使用数字 $1$ ），但要排除 $000$

For three padded digits, each position has $9$ choices if digit $1$ is forbidden, but exclude $000.$

可以按位数分类。

一位数共有 $8$ 个不含数字 $1$ 。

两位数共有 $8 \cdot 9 = 72$ 个不含数字 $1$ 。

三位数共有 $8 \cdot 9 \cdot 9 = 648$ 个不含数字 $1$ 。

因此不含数字 $1$ 的数共有 $8 + 72 + 648 = 728$ 个。

所以正确答案是 D。

We can case on the number of digits.

There are $8$ one digit numbers excluding $1.$

There are $8 \cdot 9 = 72$ two digit numbers that lack the digit $1.$

There are $8 \cdot 9 \cdot 9 = 648$ three digit numbers that do not include $1.$

This yields a total of $8 + 72 + 648 = 728$ numbers that do not contain the digit $1.$

Thus, D is the correct answer.