Question: Books on a Shelf

Comment on Books on a Shelf

Can you clarify this why you multiply arrange subjects in this math problem? I can't understand this part!
greenlight-admin's picture

In the solution, the first step is to select the order in which the SUBJECT blocks appear.
There are 6 ways to do this:
Math - Art - History
Math - History - Art
History - Art - Math
History - Math - Art
Art - History - Math
Art - Math - History

If we don't include this step, then we aren't account for the 6 possibilities above.

My issue is why are we now using factorals on each arrangement. If you said there were 3 ways to arrange the maths books,2 ways to arrange the history, etc, I would understand. But I dont understand why it is 3!. doesnt that mean 3x2x1?
greenlight-admin's picture

In the lesson for factorials (, we learned that we can arrange n unique objects in n! ways.

So, for example, we can arrange the 3 math books in 3! ways.
Likewise, we can arrange the 2 art books in 2! ways.

Does that help?

It does now! Thank you.

How would the result change if we didn't have to group the books according to subjects?
greenlight-admin's picture

If the 7 books could be arranged in any order, then we could accomplish this task in 7! ways.

Why we didnt use the counting principle here?
greenlight-admin's picture

If you're referring to the video question above, we did use the Fundamental Counting Principle (FCP) to solve this question.


why did we start with women when men is more restrictive?
greenlight-admin's picture

Question link:

Good question!
Even though there's a restriction on the men, that same restriction also applies (indirectly) to the women.
For example, if no 2 men can stand together, that also means that all 4 women can't stand together.

Does that help?


Is there a way to identify such ambiguous restrictions?
greenlight-admin's picture

Not really.
In most cases, a restriction on one part of the question will spill over into other parts.

Consider this question:
A, B, C, D and E must stand in a row. B must stand in the MIDDLE.
In how many different ways can we arrange all 5 people?

The restriction seems to apply to person B. However, the restriction also places limits on the other people as well, since they cannot stand in the middle position.

If you come across a question in which you're unsure how to deal with a restriction, just let me know, and we'll tackle the question together.


Hi Brent, I wanted to tackle the last problem you mentioned.
In here we have this _ _ B _ _
Does this mean we cna accomplish this in 4*3*2*1 ways = 24 ways? Thanks
greenlight-admin's picture

That's correct, Carla.

That is, we can seat person B in 1 way (in the middle).
We can seat person A in 4 ways (since there are now 4 chairs remaining)
We can seat person C in 3 ways (since there are now 3 chairs remaining)
We can seat person D in 2 ways (since there are now 2 chairs remaining)
We can seat person E in 1 way (since there's now 1 chair remaining)

So, the total number of ways to seat all five people equals = (1)(4)(3)(2)(1)
= 24

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