Question

In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary’s in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are five letters in all, and the boss delivers them in the order $1,$ $2,$ $3,$ $4,$ $5,$ which of the following could not be the order in which the secretary types them?

$1,$ $2,$ $3,$ $4,$ $5$

$2,$ $4,$ $3,$ $5,$ $1$

$3,$ $2,$ $4,$ $1,$ $5$

$4,$ $5,$ $2,$ $3,$ $1$

$5,$ $4,$ $3,$ $2,$ $1$

Answer: D

Concepts:stack order permutations logical feasibility

Difficulty rating: 1590

Small Hint:

Letters still in the pile always remain in increasing order from bottom to top

Big Hint:

After letter $4$ is typed, examine which smaller letter must be on top before $2$ can be typed

Solution:

To type $4$ first in choice D, letters $1,2,3,4$ must all have been delivered, leaving $3$ on top after $4$ is removed. Letter $5$ can then be delivered and typed, but $3$ is still above $2,$ so $2$ cannot be typed next. Thus $4,5,2,3,1$ is impossible. Each other listed order can be produced by interleaving deliveries and typings.

Therefore the correct answer is D.

1987 AMC 12 Problem 12

Answer: D

Difficulty rating: 1590

Small Hint:

Big Hint:

Solution:

Problem 12 in Other Years