Problem Solving

alex.gellatly wrote:Each signal that a certain ship can make is comprised of 3 different flags hanging vertically in a particular order. How many unique signals can be made by using 4 different flags?
10
12
20
24
36
[spoiler]I made the classic mistake of using C and not P.... now I understand. But does anyone have some advice on how to avoid this, or similar problems?[/spoiler]
Thanks

The "Does order matter?" question can sometimes be problematic.
I prefer to begin most counting questions by asking, "Can I take the entire task (create a signal) and break it into stages? (this is often called the slot method)

In this case, we can break the task into 3 stages:
Stage 1: Select the top flag
Stage 2: Select the middle flag
Stage 3: Select the bottom flag

At this point, we'll ask question that is analogous to the "Does order matter?" question. My question is, "Does the outcome of each stage differ from the outcomes of the other stages?"

For example, is selecting a blue flag for stage 1 different from selecting a blue flag for stage 2?
Well, according to the wording, these are, indeed, different outcomes. As such, we can continue with the slot method. If the stages are not different, then we'll have to come up with a new approach (which may or may not include combinations).

Now, let's continue.
Stage 1: There are 4 flags to choose from, so this stage can be accomplished in 4 ways.
Stage 2: Once we have completed stage 1, there are 3 flags remaining. So, this stage can be accomplished in 3 ways.
Stage 3: Once we have completed stages 1 and 2, there are 2 flags remaining. So, this stage can be accomplished in 2 ways.

So, by the Fundamental Counting Principle (FCP), the total number of ways to accomplish all 3 stages (and create a signal) is equal to 4 x 3 x 2 = 24

If you'd like more information on the FCP, you watch the following free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

Cheers,
Brent