Which one of the following is a possible matching of employees with the offices they select?

Irina on October 11, 2019

@odonnell,

This is an unusual and quite challenging game. Let's look at the setup.

Four employees - J L P T - select from among four offices - W X Y Z. The order in which they select, from first to fourth, is to be decided by a random drawing.

Initially, this appears to be a matching game that requires us to match employees to offices, but there are many possible combinations (24 to be exact) because the only restrictions are the order of selection and the order of preference. Each employee has ranked the offices from 1 to 4 as follows:

J: Y X Z W
L: X Z W Y
P: Y Z X W
T: X Y Z W

The following rules apply:
(1) Each employee selects an office that has not been selected previously.
(2) Each employee selects only one office.

These rules tell us that no employee shares an office, so we must assign four employees to four offices, and the first employee gets to pick from 4 offices, the second from the remaining three, the 3d from 2, and the fourth employee gets whatever office is left.

(3) Each employee selects the office that he or she ranks highest among the unselected offices.
This rule tells us that each employee always picks their first choice - so J would pick Y, L pick X etc unless their first choice has been picked by someone else, then they go to their second choice and so on. Let's consider one scenario to illustrate how it works in practice. Let's say the random drawing resulted in the following order: J L P T.

Under this scenario, J gets his first choice - Y, L gets his first choice - X, P can no longer get his first choice because J has already picked it, so he gets his second choice - Z, and T is left with W

J: Y
L: X
P: Z
T: W

Since X or Y is everyone's first choice, we can tell that whoever goes first, will definitely get X or Y office, a second person would get X/ Y/ or Z office, and only the 3d or the fourth person could get a W office.

The question asks us which of the following is a possible matching of employees with the offices they select?

(A) J: W, L:Y, P: X, T: Z

Incorrect. In this scenario, no one got their first choice, and we know that en employee that gets to pick first definitely gets their first choice according to the rules, thus we can eliminate (A).

(B) J:Z, L:X, P:W, T:Y

Correct. In this scenario, L goes first and gets his first choice (X), T goes second and gets his second choice (Y) since X is already taken, J goes third and gets his third choice (Z) since Y/X are taken, and P goes last and gets the remaining office - W.

(C) J:X, L:W, P:Z, T:Y

Incorrect. In this scenario, no one gets their first choice, which violates the rules.

(D) J: Y, L:W, P:X, T:Z

Incorrect. In this scenario, J goes first and gets his first choice (Y), but then no one else gets their first or second choice, which again contradicts the rules because the second person to pick an office always gets either their first (if still available) or their second choice.

(E) J:Y, L:Z, P:X, T:W

Incorrect. In this scenario, J gets his first choice, but then if L goes second, he should get his first choice (X) since it is still available, not his second choice (Z). If P goes second, then he should get his second choice (Z), not his third choice (X). Since this order is inconsistent with the rules, (E) is incorrect as well.

Let me know if this helps and if you have any further questions.

nikkim on May 22, 2020

This explanation was really great!

nikkim on May 22, 2020

This explanation was really great!

BenMingov on May 31, 2020

Hi Nikkim, we are glad! Keep up the good work!

Langston on April 11, 2021

Amazing explanation! Thanks a lot!