1995 AMC 12 Problem 11

Attempt Problem 11 of the 1995 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1995 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

11.

How many base

10

four-digit numbers,

N=\underline{a\,b\,c\,d},

satisfy all three of the following conditions?

(i)

4{,}000\le N\lt6{,}000;

(ii)

N

is a multiple of

5;

(iii)

3\le b\lt c\le6.

10

18

24

36

48

Determine the possible digits

a

and

d

from conditions (i) and (ii)

Choose two distinct digits from

3,

4,

5,

6

for

b\lt c

The thousands digit

a

4

5,

and divisibility by

5

makes

d

either

0

5.

The pair

(b,c)

is obtained by choosing two of

3,

4,

5,

6

in increasing order, giving

\binom42=6

pairs. Thus the count is

2\cdot2\cdot6=24,

and the correct answer is C.

Problem 11 in other years