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Problem 191

Combinatorics Difficulty 1.3 Multiple choice CEMC Fermat · Canada · 2014

In a group of five friends:

Amy is taller than Carla.
Dan is shorter than Eric but taller than Bob.
Eric is shorter than Carla.

Who is the shortest?

Pick one

Official solutions — 2

Solution 1

We use $A,B,C,D,E$ to represent Amy, Bob, Carla, Dan, and Eric, respectively.

We use the greater than symbol ( $\text{gt;}$ ) to represent “is taller than" and the less than symbol ( $\text{lt;}$ ) to represent “is shorter than".

From the first bullet, $\text{A gt;C}$ .

From the second bullet, $\text{D lt;E}$ and $\text{D gt;B}$ so $\text{E gt;D gt;B}$ .

From the third bullet, $\text{E lt;C}$ or $\text{C gt;E}$ .

Since $\text{A gt;C}$ and $\text{C gt;E}$ and $\text{E gt;D gt;B}$ , then $\text{A gt;C gt;E gt;D gt;B}$ , which means that Bob is the shortest.

Solution 2

We use $A,B,C,D,E$ to represent Amy, Bob, Carl, Dan, and Eric, respectively.

We use the greater than symbol ( $\text{gt;}$ ) to represent “is taller than" and the less than symbol ( $\text{lt;}$ ) to represent “is shorter than".

From the first bullet, $\text{A gt;C}$ .

From the second bullet, $\text{D lt;E}$ and $\text{D gt;B}$ so $\text{E gt;D gt;B}$ .

From the third bullet, $\text{E lt;C}$ or $\text{C gt;E}$ .

Since $\text{A gt;C}$ and $\text{C gt;E}$ and $\text{E gt;D gt;B}$ , then $\text{A gt;C gt;E gt;D gt;B}$ , which means that Bob is the shortest.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.