Merry Christmas, Magooshers!

Our gift to you? The answer to this week’s Magoosh Brain Twister! (Just what you wanted, am I right?)

Before we give you the answer, take a moment to refresh your memory.

## Question

Melilla, Tasha, and Chester plan to fly on a jetliner that has seven seats per row. The middle row consists of three seats, which are separated on each side by an aisle and two seats. The website allows users to choose specific seats, but a computer glitch only assigns rows, not the specific seat within that row. Assuming Melilla plans to book three tickets in the same row, what is the probability that Chester will sit next to Melilla, without any aisle between them?

- 3/35
- 1/5
- 4/21
- 6/35
- 1/7

## Answer and Explanation

To set up this question, you might want to draw little dashes representing the seats (don’t forget the aisles!), so you can better visualize the problem:

__ __ __ __ __ __ __

The quickest way to think of this problem is as a combinations question. But first, notice that Tasha is not important to the final answer. We essentially are asking how many different ways Melilla and Chester can sit so that they are next to each other (discounting for order, i.e., is Melilla on Chester’s left or right). We will divide this number by the total number of ways Melilla and Chester can sit, discounting, again, for order.

Melilla and Chester could sit in either of two pairs of seats on the edge, or they can sit in the middle row two different ways: one on the left end of the middle row and one in the exact middle, or one on the right end of the middle row and one exactly in the middle. This gives us four different ways:

4/7C2 = 4/21 Answer: (C).

Enjoy your holiday!

For finding out the total number of outcomes, i performed 7C3. Why is that wrong? Question says that Melilla books 3 seats in the same row – then why did we perform 7C2 while solving?

Hi Preyansh,

Even though three people are mentioned in this question, we really only care about two of them: Melilla and Chester. According to this question, we don’t really care where Tasha sits. We only care about whether or not Melilla and Chester sit together. This is why we use 7C2: out of 7 seats, we need to find out the probability that TWO people sit together 🙂

How would this question be distinguished from a permutation question (such as chester to the right of melilla is different from melilla to the right of chester)? How would thw question have been worded differently to make this the case?

Hi Vivek 🙂

Since there are more seats than people, this situation is a bit more complicated than other combination/permutation questions you’ll see on the exam. That said, to have a question that uses permutations, order needs to matter. So, one way this situation could be made into a permutations question would be by asking in how many different arrangements could the three friends be seated. In this case, it would matter who is seating next to whom.

Hope this helps! 🙂

Hi Chris,

the arrangement of Melilla and Chester should also matter. right?

I found out 8 possibilities instead of 4.

M C _ _ _ _ _

C M _ _ _ _ _

_ _ M C _ _ _

_ _ C M _ _ _

_ _ _ M C _ _

_ _ _ C M _ _

_ _ _ _ _ M C

_ _ _ _ _ C M

Can you tell me where I am going wrong?

Hi there 🙂

Happy to clarify! While you’re right that M could sit on C’s left or right, in this problem, this doesn’t matter. That’s because what we’re really finding is the way to have two of the three seats assigned next to each other. At that point, M can choose to sit to the left or right of C. As long as the two are next to each other, we’re all set 🙂

When writing arrangements, then, we can consider M and C indistinguishable. Let’s call them both X, for example, so that the possible arrangements with both Xs next to each other are:

XX ___ __

__ XX_ __

__ _XX __

__ ___ XX

As you can see, there are four ways to have two seats next to each other. Again, we don’t consider whether M is to the left or right side of C, because the two are free to choose their seat, as long as they are seating next to each other.

I hope this clears up your doubts! If not, let us know 🙂

That was extremely mind-boggling to say the least! I don’t get how you determined the ways that they could be seated in the middle row. When I worked it out I got 5 possible ways. I feel like I know how to solve the problems but I get rattled by the extraneous information and tend to over think the questions.

Hi Peter,

I agree that the problem has many moving parts! Simplification is key–that’s why drawing out dashes to represent the seats and using initials for the names will make things easier. Speaking of which, here is how it works out for the three middle seats:

M C _

C M _

_ C M

_ M C

These are the only four possibilities. We can’t include M _ C or C _ M, because then there is a seat between them.

Hope that helps!

is this kind of math can be appeared in the real test???

No, this is slightly more difficult. You could get something similar, but not as convoluted.

Sorry if the problem scared you 🙂