Suppose n children are going to sit, in any order, on a row of n chair

Question

Suppose n children are going to sit, in any order, on a row of n chairs. What is the value of n?

(1) In this row, there are 5040 possible arrangements of the n children.
(2) If one child leaves, and one chair is removed from the row, there would be 4320 fewer arrangements of the remaining children.

Bunuel · Accepted Answer

MAGOOSH OFFICIAL SOLUTION:

This is a question about  permutations . Because this is Data Sufficiency, we don’t have to bother with the details of calculating n! —- it’s enough to know that if we know the value of n, we can figure out the value of n!, and vice versa!

Statement #1: this gives us n! = 5040 — from this we could figure out n. Statement #1, alone and by itself, is sufficient.

Statement #2: this tells us, essentially, n! – (n – 1)! = 4320. Well, if we went through list of  factorials , there could be only one pair that is separated by exactly 4320. We don’t have to know what it is — it’s enough to know that this fact determines a unique value of n. Statement #2, alone and by itself, is sufficient.

Answer = D.

(BTW, n = 7 if you’re curious!)

Psiva00734 · Answer

statement 1
If n children sits in nchairs then arrangements is n!

Total arrangements is 5040. it equals to 7!.

It is sufficient.

Statement 2
By removing one children from n is( n-1) and one chair from n row is( n-1) than arrangements is n! - (n-1)!.
Arrangement is 4320 left. it means 7! - 6!.
5040 - 720= 4320.
It is also sufficient.

so answer is D

EMPOWERgmatRichC · Answer

Hi All,

There are going to be a few questions on Test Day that require the use of factorials. While it's not necessary to have these calculations memorized, if you DO have them memorized, then you'll likely be able to move much faster on these handful of questions.

2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040

You're not likely to have to deal with much higher than 7! as a 'solo' calculation. In a question such as this one, this knowledge can help you to quickly work through the prompt.

GMAT assassins aren't born, they're made,
Rich

TheKingInTheNorth · Answer

Suppose n children are going to sit, in any order, on a row of n chairs. What is the value of n?

(1) In this row, there are 5040 possible arrangements of the n children.
(2) If one child leaves, and one chair is removed from the row, there would be 4320 fewer arrangements of the remaining children.

Statement 1 . Sufficient 
because n children can be arranged in n chairs in n! ( n factorials ways )
and 7! =5040

Statement : 2 Sufficient 
Because 7! = 5040 
and if we remove one person and one chair then there would be 4320 scenario .

6!=720 
7! - 6! = 4320

mynguyen62 · Answer

Sorry if I'm asking naive questions but all of the above members have no difficulties in pointing out that 7! = 5040, so that they could easily reach the answer.

To me, 5040 seems alien. I can't tell 5040 is a factorial at a glance.

Okay, I see someone recommending that it is useful to memorize the factorials from 1! to 7!. That may be the method to solve this particular question.

But what if the question gives random numbers, which we don't know for sure if they are factorials or not? For example, statement (1): n! = 12345, statement (2): n! - (n-1)! = 1234.

Thanks for any help,

EMPOWERgmatRichC · Answer

Hi mynguyen62,

As a subject, factorials are relatively rare on the GMAT (you'll likely see just one question on the subject on Test Day). That having been said, nothing about factorials is "random", so there cannot be a Quant question involving factorials that uses random numbers. As such, if you're told that N! = a specific number, then there is just ONE possible value for N (and the same can be said for the difference between two Factorials). Knowing that 7! = 5040 might make it easier to 'process' the information in this question, but it's not necessary to know that IF you realize that a different "N" leads to a different/unique result for N!

GMAT assassins aren't born, they're made,
Rich

Sachin07 · Answer

Statement 1 . Sufficient 
there are 5040 possible arrangements of the n children And because n children can be arranged in n chairs in n!
thus n!=5040
Using prime factorisation I found n=7

Statement : 2 Sufficient 
and if we remove one person and one chair then there would be 4320 scenario
n!-(n- 1)!=4320
Taking (n-1) common , (n-1).(n-1)!=4320
Using prime factorisation u can find (n-1) to be 6
6*6! = 4320

And D

GMATBusters · Answer

Bunuel ,  chetan2u ,  EMPOWERgmatRichC ,  VeritasPrepKarishma

if  n!-(n-1)!  = K (a constant integer)

Can the value of n be uniquely determined ?

KarishmaB · Answer

Note the pattern:
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720

The difference between consecutive factorials keeps increasing (obviously since every time you are multiplying it by the next greater number).
So there will be only one pair of consecutive values for any given constant.

Say K = 18. Then n = 4
Say K = 600. Then n = 6
and so on...

GMATBusters · Answer

Thank you for giving insights...

Note the pattern:
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720

The difference between consecutive factorials keeps increasing (obviously since every time you are multiplying it by the next greater number).
So there will be only one pair of consecutive values for any given constant.

Say K = 18. Then n = 4
Say K = 600. Then n = 6
and so on...

ColumbiaEMBA · Answer

Suppose n children are going to sit, in any order, on a row of n chairs. What is the value of n?

(1) In this row, there are 5040 possible arrangements of the n children. 
(2) If one child leaves, and one chair is removed from the row, there would be 4320 fewer arrangements of the remaining children.

1) Would GMAT give 5041 as an answer? Nope, never seen it. Given it's a straight number, this will be sufficient. 
2) Same logic applies. Would the difference of n! - (n-1)! be some random number? Nope; thus, this will be sufficient.

Without solving, I would wager on D.

BeePen · Answer

Explain u posted is helpful ! but do not understand why B can be right ? How can u calculate on exam with no calc?

GMATBusters · Answer

BeePen , please see the below explanation by veritasprepkarishma

if  n!-(n-1)!  = K (a constant integer), n can be uniquely found

Note the pattern:
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720

The difference between consecutive factorials keeps increasing (obviously since every time you are multiplying it by the next greater number).

So there will be only one pair of consecutive values for any given constant .

Say K = 18. Then n = 4
Say K = 600. Then n = 6
and so on...

adkikani · Answer

Bunuel   niks18   chetan2u   pushpitkc

Can you please explain St 2: How we interpret : n! – (n – 1)! = 4320
when we have same no of children and chair and start and now we removed each of one chair and one children.

pushpitkc · Answer

Hi  adkikani

The 'n' children can be arranged in 'n' chairs is n! ways
After 1 chair is removed, the 'n-1' children can be arranged in (n-1)! ways

Statement 2 says that if one child leaves and one chair is removed

To understand this clearly, let's consider n = 3.

So, how can we seat 3 children in 3 chairs -
3 ways of seating one of the 3 children in the first chair
2 ways of seating one of the 2 children in the second chair
1 ways of seating one child in the third chair (3*2*1), which is 3!

Similarly, we can seat 2 children in 2 chairs as (2*1), which is 2!

Therefore, the total number of ways must be n! - (n-1)!

Hope this helps you!

IanStewart · Answer

Yes. If you first look at, say, 8!- 7!, you can factor out the "7!"

8! - 7! = (8)(7!) - 7! = (7!)(8 - 1) = (7)(7!)

and similarly, you can factor (n-1)! from n! - (n-1)!

n! - (n-1)! = (n)(n-1)! - (n-1)! = (n-1)! (n - 1)

Notice the larger n is, the larger (n-1)! (n-1) will be, so if you know it equals some fixed value K, there can only be one value of n that could work.

I'd be curious if anyone can point to a single official problem where it would be useful to have memorized the value of 7! in advance. I can't imagine an official GMAT question where that would be helpful.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Suppose n children are going to sit, in any order, on a row of n chair

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

Suppose n children are going to sit, in any order, on a row of n chair

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards