Lesson: Factorial Notation

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

Ha! Guilty as charged!
Sorry about being Canadian :-)

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?

Question link: http://gre.myprepclub.com/forum/four-women-and-three-men-must-be-seated-...

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.

Question link: https://gre.myprepclub.com/forum/four-women-and-three-men-must-be-seated...

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

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

Question link: https://gre.myprepclub.com/forum/four-women-and-three-men-must-be-seated...
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

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