# Lesson: Factorial Notation

## Comment on Factorial Notation

### I'm still not sure what you

I'm still not sure what you mean by "stages," logically. Can you define what that means, perhaps by example? I can see that each step is a stage, but I'm not sure if that's the gist.

### For the purposes of these

For the purposes of these videos, "stages" and "steps" can be used interchangeably. Sorry for any confusion.

### "Sorry for any confusion."

"Sorry for any confusion."
If it helps I wasn't confused by stages.

### Ha! Guilty as charged!

Ha! Guilty as charged!

### Am always confused with the

Am always confused with the the(n - 1) concept in counting. In this case n x (n-1)x(n-2)x..., can you please explain with a number example. Thanks

### Sure thing.

Sure thing.

Let's start with: 5! = 5 x 4 x 3 x 2 x 1

We could also say: 5! = 5 x (a number that's 1 LESS THAN 5) x (a number that's 2 LESS THAN 5) x (a number that's 3 LESS THAN 5) x (a number that's 4 LESS THAN 5)

Or we might write: 5! = 5 x (5 - 1) x (5 - 2) x (5 - 3) x (5 - 4)

Likewise, n! = n x (a number that's 1 LESS THAN n) x (a number that's 2 LESS THAN n) x (a number that's 3 LESS THAN n) x (a number that's 4 LESS THAN n) x (a number that's 5 LESS THAN n) x . . .

Or we might write: n! = n x (n - 1) x (n - 2) x (n - 3) x (n - 4) x . . . . all the way down to . . . 2 x 1

Does that help?

### http://gre.myprepclub.com/forum

http://gre.myprepclub.com/forum/four-women-and-three-men-must-be-seated-in-a-row-for-a-group-5111.html

I thought there will be 7 chairs in a row, on alternative chairs women will sitting W_W_W_W, left with 3 chairs in between.
W can sit in 24 different ways, M can sit 3! (6) different ways. totally 144 ways.

how can we assume this pattern _W_W_W_W_ ?

Be careful. The question does NOT require us to keep the women separated. The question requires us to ensure that no two men are seated next to each other.

So, as you suggest, one possible arrangement is W_W_W_W, where we seat the 3 men in the 3 spaces.

HOWEVER, that's not the only scenario in which the men are separated.

Here's another _WW_W_ (place the 3 men in the 3 spaces)
And another _W_WW_ (place the 3 men in the 3 spaces)
And another _WW_W (place the 3 men in the 3 spaces)
.
.
.
etc

My solution considers all of the possible arrangements.

### Hi Brent... thanks for

Hi Brent... thanks for everything, I want to find out whether the restriction method could be use to solve the question in the reinforcement activities.

Hi Angel,

That approach would take a VERY VERY long time.
The reason is that there are MANY MANY MANY different ways to break the rule that says "no two men can sit next to each other."

If we let the W, W, W, W, m, m, m represent the 4 women and 3 men, here are just a few of the ways that we can break the above rule:
- W.m.m.W.W.m.W
- W.m.W.W.W.m.m
- m.m.m.W.W.W.W
- W.m.W.m.m.W.W
- W.W.m.m.m.W.W
etc

There are MANY more configurations that break the rule.
So, as you can see, this approach is extremely time consuming.

Does that help?

Cheers,
Brent

I understand your solution, but I'm wondering that the question does not mention that we could have blank spaces between the people. If we assume to have 5 places for 3 men then there will be two empty chairs next to the women.

### Good question.

Good question.
Once everyone has been seated, we just remove the two empty chairs and, voila, we have a unique arrangement.
Does that help?

Cheers,
Brent

### That Greenlight practice

That Greenlight practice question is ridiculously hard...too hard for GRE standards. Definitely one to skip on first damn sight.

I agree that it's a super hard question.
If it's too hard to be an official GRE question, it's just barely too hard (171-level :-).

That said, as you can see from the various Solutions, there's more than one way to solve this question.
So, although my strategy might be a little out of the box, there are other strategies that one might and reasonably expect high-scoring students to solve.

Cheers,
Brent

### How would factorial notation

How would factorial notation of fractions work like (6/2)! or (2/6)!?

### Factorial notation doesn't

Factorial notation doesn't apply to non-integers.

Excerpt from the Official Guide:
n! is the product of all positive integers less than or equal to n, where n is any POSITIVE INTEGER and, as a special definition, 0! = 1.

ASIDE: In the case of (6/2)!, 6/2 simplifies to be an integer.
So we get: (6/2)! = 3! = (3)(2)(1) = 6

Cheers,
Brent

### Hi Brent,

Hi Brent,

Regarding this practice question
https://gre.myprepclub.com/forum/four-women-and-three-men-must-be-seated-in-a-row-for-a-group-5111.html

I'm sorry to bother you with yet another question on this matter.

"For each arrangement of 4 women, there are 5 spaces where the 3 men can be placed."

Would you please explain what you mean? (I have read all of your answers above and on the forum, but I am still unclear.)

More specifically, what does "each arrangement of 4 women" mean?

When I wrote out what I thought was "EACH arrangement of 4 women," I did not see 5 spaces. I saw 3.

For the life of me, I cannot figure out what that sentence means, let alone why it should be self-evident.

I solved by listing* the possibilities for "each arrangement of 4 women." Maybe I am defining the latter incorrectly.

I'll call women "F" and men "M." The letters "W" and "M" are hard to distinguish.

• Here are my arrangements (10 cases):

1. F F M F M F M
2. M F M F M F F
3. F M F F M F M
4. M F M F F M F
5. M F F M F M F
6. F M F M F F M
7. M F F F M F M
8. M F M F F F M
9. F M F M F M F
10 M F F M F F M

• In each case, the # of ways to seat the 4 women equals
4*3*2*1 = 4!

• In each case, the # of ways to seat the 3 men equals
3*2*1 = 3!

• So for EACH case, the total number of possible arrangements is 4!*3!= 144

• There are 10 such cases.
So, 144*10 = 1,440 (correct)

I *think* that I satisfied the part of your sentence that states, "For each arrangement of 4 women ..."

After that part, there is just one problem from where I am standing, namely:

For each arrangement of 4 women that I can see in *my* list, there are not "5 spaces where the 3 men can be placed."
There are, rather, 3 such spaces where the 3 men can be placed.

I played around with my 10 arrangements to see whether I could sensibly get 5 spaces for the 3 men.
No luck.

Am I missing something obvious? I'll be happy to take it on faith that I just need to add two seemingly extra spaces in questions with this setup and its attendant conditions.

*Doing so is not too horrible. Eight of 10 are mirror images.
Find one arrangement. Rewrite or visualize its letters written exactly backwards. Check to see whether the result is a new arrangement.

For #1 through #8, the letters in each even numbered answer are written in the reverse order of the letters in the immediately preceding odd numbered answer,
Arrangements 9 and 10 are themselves palindromic.

Here's what that sentence means:

Take 9 chairs: _ _ _ _ _ _ _ _ _
Seat the 4 women in 4 chairs as follows: _W_W_W_W_. (this step can be performed in 4! ways)

There are still 5 empty chairs (for the men).
Notice that, by the way I've already seated the women, we will be able to obey the condition that no two men sit next to each other.

At this point, we must seat the 3 men.
There are 5 chairs. So, once we've seated the 3 men, we can just throw away to 2 unused chairs, and now we have an arrangement of 4 women and 3 men.
We can seat the 3 men in (5)(4)(3) ways.

Does that help?

Your solution is perfect. It shouldn't be compared to my solution, because they're completely different solutions.

### For this question

For this question

https://gre.myprepclub.com/forum/four-women-and-three-men-must-be-seated-in-a-row-for-a-group-5111.html

If possible, could you provide a solution where we think of first sitting the boys. I started with that since I believe that is the most restrictive stage, but working at it I was a bit lost. Thanks

That strategy would prove very difficult.
Here are all 10 possible cases:
1. W_W_W_W
2. _WW_W_W
3. WW_W_W_
4. _W_WW_W
5. W_WW_W_
6. _W_W_WW
7. W_W_WW_
8. _WWW_W_
9. _W_WWW_
10. _WW_WW_

We would have to treat each case separately.

Keep in mind that this is a very tricky question. It may even be beyond the scope of the GRE.