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Combinatorics Difficulty 2.4 Junior Find the answer Canada

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

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 $F$ .

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