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Worked solution

Multiply the choices for two successive road stages

There are 2 roads connecting City A and City B, and 3 roads connecting City B and City C. There is no direct road that links City A to City C. How many ways are there to travel from City A to City C?

CJC · Lecture · Example 2

Read the two stages from the diagram

Number of ways from A to B=2; number from B to C=3. A complete route chooses one road in each stage, travelling through B.

How did this part go?

Pair every first-stage choice with every continuation

Each of the 2 choices can be followed by any of the 3 roads. Total number of ways=2×3=62\times3=6. Label the roads 1,2 and 1,2,3 to list the six pairs: (1,1),(1,2),(1,3),(2,1),(2,2),(2,3). The list supplements the source multiplication.

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Source diagram: cities A,B,C in sequence, with two directed roads A to B, three B to C and no direct A to C. The six ordered road pairs are listed, showing the multiplication principle.
Source diagram: cities A,B,C in sequence, with two directed roads A to B, three B to C and no direct A to C. The six ordered road pairs are listed, showing the multiplication principle.
Source diagram: cities A,B,C in sequence, with two directed roads A to B, three B to C and no direct A to C. The six ordered road pairs are listed, showing the multiplication principle.

Common mistakes

These are stages of one journey, not alternatives. The model counts direct A→B→C journeys, without extra loops or backtracking.

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