# 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://greprepclub.com/forum

http://greprepclub.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