AMC 8 2002 Problem 19

AMC8 2002 年第 19 题

基本计数原理★★★☆☆

How many whole numbers between 99 and 999 contain exactly one 0?

99 和 999 之间有多少整数恰好包含一个 0?

A.72
B.90
C.144
D.162
E.180

正确答案 · Correct Answer

解析 · Solution

Solution 1

Numbers with exactly one zero have the form

\overline{a0b}

\overline{ab0}

, where the

a,b \neq 0

. There are

(9\cdot1\cdot9)+(9\cdot9\cdot1) = 81+81 = 162

such numbers, hence our answer is

\fbox{D}

. (Whole numbers between 99 and 999)-(Whole numbers which do not contain exactly one 0)= (How many whole numbers between 99 and 999 contain exactly one 0). 1) How many whole numbers are between 99 and 999, 999-99=900, so there are 900 numbers between 99 and 999. 2a) How many whole numbers in this range do not contain the digit 0, there are 10-1=9 possible digits for each digit in this three digit number,

9^3=729

. So there are 729 numbers in this range which do not contain the digit 0. 2b) How many whole numbers in this range contain the digit 0 2 times, there are 10-1=9 possible digits for the first digit and the other two digits have to be the digit 0. So there are 9 of these numbers. So there are 900-(729+9) whole numbers between 99 and 999 that contain exactly one 0. Simplifying the expression we get that there are 162 whole numbers between 99 and 999 that contain exactly one 0. So the answer is

\fbox{D}

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