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Problem 284

Combinatorics Difficulty 2.4 Multiple choice CEMC Pascal · Canada · 2022

On the map shown, there are a number of routes from Mathville to
Algebratown.

Each route must travel along the roads in the direction marked by the
arrows. The total number of routes from Mathville to Algebratown is

Pick one

Official solution

We label Mathville as $M$ ,
Algebratown as $A$ , and the other
intersection points of roads as shown.

[[IMAGE0]]

There is 1 route from $M$ to each
of $C$ and $B$ : $M $\to$
C $and$ M $\to$ B$.

There are 3 routes to $D$ : $M \to D$ and $M
$\to C \to$ D $and$ M $\to B \to$
D$.

This means that there are 4 routes to $F$ : $M \to C \to F \qquad M \to D \to F \qquad M \to C \to D \to F \qquad M \to B \to D \to F$ Similarly, there are 4 routes to $E$ : $M \to B \to E \qquad M \to D \to E \qquad M \to C \to D \to E \qquad M \to B \to D \to E$ Finally, there are $4+4=8$ routes to $A$ , since every route comes through either
$E$ or $F$ , no route goes through both $E$ and $F$ , and there are 4 routes to each of
$E$ and $F$ .

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