2006 AMC 12B Problem 9

Question

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

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

For each even units digit

c,

count the ways to choose two increasing digits below

c

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.

Problem 9 in other years