Question: Books on a Shelf

Permutations, the "N! method" and the FCP are all members of the same family. They all require us to break a task into stages.

For example, if we want to arrange 6 objects in a row, we can use the FCP and see that the first position can have 6 different objects, the second position can have 5 different objects, the third position can have 4 different objects, etc. And when we multiply the stages, we get (6)(5)(4)(3)(2)(1).

Or we can just use the N! formula, which says N objects can be arranged in a row in N! ways.

We essentially used the FCP to derive the N! formula, so either method is fine.

Permutations fall into the same category, except I'm not a big fan of teaching permutations since there are very few true permutation questions on the GMAT, and since we can use the FCP to solve permutation questions anyway.

More on this here: https://www.gmatprepnow.com/articles/combinations-and-non-combinations-%...

We can arrange n objects in n! ways. So, we can arrange the 3 subjects (math, art and history) in 3! ways (= 6 ways)

By the way, here are the 6 ways:

1) math, art, history
2) math, history, art
3) history, art, math
4) history, math, art
5) art, history, math
6) art, math, history

Cheers,
Brent

Good question jpshow.

The difference here is that we're told the 3 math books are all DIFFERENT. Likewise, the 2 art books are different and the 2 history books are different.

The MISSISSIPPI Rule involves arranging a group of items in which some of the items are IDENTICAL (like the letters S, S, S and S)

Cheers,
Brent

That's a good start, but what do you do if the first book isn't a Math book?

Question link: https://gmatclub.com/forum/team-a-and-team-b-are-competing-against-each-...

You've suggested a valid approach, except for the last part (3!).

Here's how that approach would look:

STAGE 1: Arrange the 3 males from left to right
We can complete this stage in 3! ways (= 6 ways)

STAGE 2: Arrange the 3 females from left to right
We can complete this stage in 3! ways (= 6 ways)

STAGE 3: Combine the two groups so that we have male-female-male-female-male-female.
We can complete this stage in 1 way

So, the total number of arrangements = (6)(6)(1) = 36

NOTE: Some students will not see that STAGE 1 can be completed in only 1 way.
Let me try to convince them of this.

Let's say that, in STAGE 1, we arranged the 3 males as follows:
1st = Joe, 2nd = Al, and 3rd = Ed

And let's say that, in STAGE 2, we arranged the 3 females as follows:
1st = Sue, 2nd = Ann, and 3rd = Bea

In STAGE 3, we'll combine them to go in the order: male-female-male-female-male-female.
Since the 1st person is male, the 1st person will be Joe (since he's first among the males)
The 2nd person is female. So, the 2nd person will be Sue (since she's first among the females)
3rd person = Al
4th person = Ann
5th person = Ed
6th person = Bea

So, once we've completed STAGES 1 and 2, there's only one way to arrange the 6 people.

Does that help?

Cheers,
Brent

1) What if the question asked us to arrange the subjects in any order?
The question does allow for subjects to be arranged in any order (e.g., History-Math-Art or Math-History-Art or Art-History-Math or Math-Art-History, etc

2) What would be the number of ways if each subject had the same book and there were 2 books for each subject?
In this case, the answer would be 3! (6), since all we'd need to do is arrange the 3 subjects, which can be done in 3! ways. Once we've arranged the 3 subjects, we just place two (identical) books in each section.

Does that help?

Cheers
Brent

If the 7 books and be arranged in any order, then we can do sew in 7! ways (since we are arranging 7 unique objects)
That is, we have 7 options for the first book in the arrangement.
We have 6 options for the second book in the arrangement.
We have 5 options for the third book in the arrangement.
And so on