2006 AMC 12B 第 9 题

Question

有多少个三位偶数满足：从左到右读，它们的各位数字严格递增？

How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

21

34

51

72

150

个位数字是偶数；前两个较小的数字都必须小于它

The units digit is even; the two smaller digits must both be less than it

对每个偶数个位

c

数出从小于

c

的数字中选两个递增数字的方法数

For each even units digit

c,

count the ways to choose two increasing digits below

c

设三个数字为

a \lt b \lt c

，且

c

为偶数。因为

a \geq 1

，没有数字为零，并且

c \neq 2

（没有两个更小的非零数字可选）。

一旦个位数字

c

固定，任意两个小于它的不同数字都能按递增顺序唯一排列。因此每个

c

的计数是

\binom{c-1}{2}

。

对

c = 4, 6, 8

分别计数，得到

\begin{aligned} &\binom{3}{2} + \binom{5}{2} \\ &\quad {}+ \binom{7}{2} = 3 + 10 + 21 \\ &\quad = 34 \end{aligned}\text{。}

因此，正确答案是 B。

Let the digits be

a \lt b \lt c

with

c

even. Since

a \geq 1,

no digit is zero, and

c \neq 2

(there is no room for two smaller nonzero digits).

Once the units digit

c

is fixed, any two distinct digits below it can be arranged in increasing order in exactly one way. So the count for each

c

is

\binom{c-1}{2}.

For

c = 4, 6, 8

this gives

\begin{aligned} &\binom{3}{2} + \binom{5}{2} \\ &\quad {}+ \binom{7}{2} = 3 + 10 + 21 \\ &\quad = 34. \end{aligned}

Thus, the correct answer is B.

其他年份的第 9 题