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Comment on Bicycle with Optional Features
When I initially read this question, my mind draws the conclusion that 5 different ways to customize this bike equates to "5!" or (through Stages/FCP method) this = 5 x 4 x 3 x 2 x 1.
Could you please explain how one is to differentiate this question-type from other question types that sound similar to this example? (I.E: In how many ways can 5 people sit in 5 different chairs?)
Albeit the solution for this question seems very straightforward, I would never have drawn the conclusion that the stages would become 2 x 2 x 2 x x 2 = 2^5.
From what I understand, would the factorial method only work if all 5 optional features in this question were the same?
Really appreciate the help...I'm "counting" my losses with this section given its difficulty, but hoping to learn as much as I can so I can at least knockoff 500-550 level Counting Qs on test day.
Thanks as always.
Your desire to want to make 5! work for this question is very common.
As students learn some of the more basic features of the Fundamental Counting Principle (FCP), they see that many of the solutions have the format (n)(n-1)(n-2)....
So, for the above video question, we have 5 optional features, so many people want to make (5)(4)(3)(2)(1) work for them.
However, as you're using the FCP to solve a question, you must make it very clear to yourself what is occurring at each individual stage.
So, for your solution of 5 x 4 x 3 x 2 x 1, you are suggesting that stage 1 can be completed in 5 ways.
To know whether or not this step is valid, we need to know what you are doing in stage 1. I'm not sure what you're doing in stage 1, so it's hard for me to comment on this.
If you can tell me what's happening at each stage of your solution, it will help clear up any confusion.
That said, notice that, in my solution, I make it clear that, in stage 1, I'm choosing whether or not the bike will have a headlight. This stage, as we know, can be accomplished in 2 ways.
Thanks for your response
1) I translate the Question Stem to: "how many different configurations are possible if the bike can have up to 5 optional features".
2) I create blank stages/slots and ask whether
a) Is each slot unique? (YES)
b) Are there any restrictions? (NO)
3) Here is where I take the wrong turn. I begin using the following approach to full in the stages/slots:
5 stages ( _ _ _ _ _ )
In Slot #1, I know that bike can have the maximum number of features, which is 5. Therefore (5 _ _ _ _)
Now if 5 maximum features are used, I am left with 4 maximum features (5 4 _ _ _)
and so on.... until I am at (5 4 3 2 1)
The problem for me is that I do not know how to "think"/"formulate" the question I need to be asking for each stage. For instance, in this example, we need to be asking YES/NO for each stage vs Max/Min.
So to clarify, is there a way to differentiate between the factorial method and the yes/no approach (i.e.: this example) when using the Stage/FCP method?
It's still not clear to me what you are doing in stage (slot) #1. Each stage should feature some kind of clear action/direction.
To give you some examples, I went through some of my solutions on the various forums, and here are a few:
Stage 1: Select the middle digit
Stage 1: Choose one of the flavors
Stage 3: Select a competitor for the 3rd position
Stage 2: Choose a hundreds digit
Stage 3: Select a point that is on the same vertical line as the first point.
In each stage, I'm performing some kind of action/direction, and I need to determine the number of ways to accomplish that stage.
That's what's missing from your solution.
One way to help you correctly define each step (with some kind of action/direction) is to first list some possible outcomes to the question.
This is step #1 in the general strategy for solving counting questions (see https://www.gmatprepnow.com/module/gmat-counting/video/791)
So, for example, with the bike question, some possible outcomes are:
1) bike has headlight, horn, pedal clips, air pump and travel bag
2) bike has headlight, pedal clips, air pump and travel bag, but no horn
3) bike has air pump and travel bag, but no horn, no headlight, and no pedal clips.
Listing a few possible outcomes will help you define what's occurring during each stage of your solution.
I got the factorial 5! result too. Could you please tell me what I did wrong? Thanks:
I thought that the bike would have 5 "slots" to "insert" optional features:
Slot 1: we can insert one from the 5 features, hence 5 ways
Slot 2: we can insert one from the remaining 4, hence 4 ways
Slot 5: only 1 way since only 1 feature remaining
So the result would be 5x4x3x2x1
Each time we identify a step (stage), we must be very clear what that step means to the solution.
To see what I mean, let's examine your first step (stage), which is:
"Slot 1: we can insert one from the 5 features, hence 5 ways"
What exactly does this mean? What would be a possible outcome of this first step?
Based on your final answer, it sounds like one of the 5 options (headlight, pedal clips, air pump, travel bag, or horn) must go in this slot.
For slot 2, you're suggesting that one of the remaining 4 features must go in slot 2.
Then, one of the remaining 3 features must go in slot 3.
So, with your solution, one possible outcome is: pedal clips, air pump, travel bag, headlight, horn
Another possible outcome is: air pump, headlight, horn, travel bag, pedal clips
Important question: What is the difference between the two outcomes listed above?
With your solution, every outcome results in a bike with all 5 features (except the order in which those features are listed is different with each outcome).
However, the question says that a bicycle can have any number of features.
Does that help?
Brent, why cant we think of
Let's consider what each
Let's consider what each number represents in your sum 5 + 4 + 3 + 2 + 1 + 0 = 15
Your sum of 15 suggests that there are 15 different WAYS to equip a bike with optional features.
So, I'm assuming that the 5 in your sum means that there are 5 WAYS to equip a bike with all 5 features.
This, however, is not true.
There is only 1 way to equip a bike with all 5 features.
What about the 4 in the sum?
Does this mean there are 4 ways to equip a bike with 4 of the 5 features?
There are, actually, 5 ways to equip a bike with 4 features.
Let's jump down to the 1 in the sum.
Does this mean there is only 1 way to equip a bike with 1 feature?
There are, actually, 5 ways to equip a bike with 1 feature.
As you can see, it's important to consider what each value represents in your solution.