Video Transcript
There are nine ships which travel between two ports. Determine the number of ways to travel from one port to another and then back again, without using the same ship for both journeys.
Let’s begin by trying to picture what’s actually happening here. We have two ports. Let’s call them port A and port B. We can travel from port A to port B on a single ship. Let’s say we choose to travel on ship number one. Then, on our way back, we can travel on a different ship. We’re not taking the same ship for both journeys, so we might choose to travel on, say, ship two. We’re looking to work out the number of different ways of achieving this same journey. And so we’re going to recall something called the fundamental counting principle.
The fundamental counting principle or the product rule for counting tells us that the total number of outcomes for two or more events is found by multiplying the number of outcomes for each event together. And so we have two events. We have event number one, which is traveling from port A to port B, and then our second event is traveling back again. So how many ways are there for us to travel from port A to port B? The question tells us that there are nine ships to choose from. So the journey from port A to port B has nine outcomes, and those are choosing a specific ship.
Next, let’s consider what happens when we travel back again. We’re told we can’t use the same ship for both journeys, and that leaves us then no matter which ship we chose to travel out on with a possible eight ships to travel back again. The fundamental counting principle tells us that the total number of outcomes, in other words, the number of ways to complete this journey without using the same ship, is the product of these two numbers. It’s nine times eight, which is equal to 72.
There are 72 ways to travel from one port to another and then back again without using the same ship for both journeys.
