GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 21 Oct 2019, 05:43

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

5 letters have to be put in 5 different envelopes numbered 1 through 5

Author Message
TAGS:

Hide Tags

Intern
Joined: 03 Mar 2016
Posts: 15
Location: India
Schools: ISB '19
5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

Updated on: 10 Mar 2016, 00:23
1
10
00:00

Difficulty:

95% (hard)

Question Stats:

27% (02:36) correct 73% (02:47) wrong based on 89 sessions

HideShow timer Statistics

5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

Originally posted by vardhanindaram on 10 Mar 2016, 00:19.
Last edited by Bunuel on 10 Mar 2016, 00:23, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert
Joined: 02 Aug 2009
Posts: 8005
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

10 Mar 2016, 01:15
1
4
vardhanindaram wrote:
5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

This Q is based on derangement, a permutation in which all elements are in the wrong position.

Number of derangements =
$$n! (\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} + ... + (\frac{(-1)^n)}{n!})$$.. Ofcourse it can be derived but thats not required..

Since there are 5 letters and 5 envelopes:-

$$Derangements = 5! (\frac{1}{2!} - \frac{1}{3!}+ \frac{1}{4!}- \frac{1}{5!})$$
=> $$120(\frac{1}{2} - \frac{1}{6}+\frac{1}{24}- \frac{1}{120}) = 60 -20 +5 -1=44.$$
Total possible ways = 5! = 120.
therefore required prob
, $$P = \frac{44}{120} = \frac{11}{30}.$$

ans D
_________________
General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 58421
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

10 Mar 2016, 00:58
vardhanindaram wrote:
5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

Similar questions to practice:
letter-arrangements-understanding-probability-and-combinats-84912.html
micky-has-10-different-letters-and-5-different-envelopes-with-him-213801.html
5-letters-have-to-be-put-into-their-5-respective-envelopes-189501.html
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 58421
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

10 Mar 2016, 00:59
vardhanindaram wrote:
5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9704
Location: Pune, India
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

10 Mar 2016, 02:13
1
vardhanindaram wrote:
5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

Note that this question is not a GMAT type question. A GMAT type P&C question would be tricky, (perhaps fun!) but do-able in a minute or two. It would not depend on your knowledge of obscure formulas and would not require you to go through a ton of calculations.
A question which deals with 3 letters and 3 envelopes and asks you to say, find the number of ways in which at least 2 letters go in the correct envelope is far more GMAT friendly.
It would require you to realise that you cannot have exactly 2 letters going in the correct envelope so all you need to do is find the number of ways in which all 3 go in the correct envelope.

This post discusses the logic of putting letters into envelopes: http://www.veritasprep.com/blog/2011/12 ... envelopes/
_________________
Karishma
Veritas Prep GMAT Instructor

Senior Manager
Joined: 21 Aug 2016
Posts: 256
Location: India
GPA: 3.9
WE: Information Technology (Computer Software)
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

11 Jun 2017, 02:43
VeritasPrepKarishma wrote:
vardhanindaram wrote:
5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

Note that this question is not a GMAT type question. A GMAT type P&C question would be tricky, (perhaps fun!) but do-able in a minute or two. It would not depend on your knowledge of obscure formulas and would not require you to go through a ton of calculations.
A question which deals with 3 letters and 3 envelopes and asks you to say, find the number of ways in which at least 2 letters go in the correct envelope is far more GMAT friendly.
It would require you to realise that you cannot have exactly 2 letters going in the correct envelope so all you need to do is find the number of ways in which all 3 go in the correct envelope.

This post discusses the logic of putting letters into envelopes: http://www.veritasprep.com/blog/2011/12 ... envelopes/

Hi VeritasPrepKarishma,

I am not able to solve this question by example approach; please help where am I making a mistake.

Let's say there are 5 letters--L1,L2,L3,L4,L5

and there are 5 envelopes- E1,E2,E3,E4,E5

L1 can be put incorrectly in 4 ways(E2,E3,E4,E5), let's say, it is put in E3
now, L3 can also be put incorrectly in 4 ways(E1,E2,E4,E5), lets say it is put in E2.

now, L2 can be put incorrectly in 3 ways(E1,E4,E5), let's say it is put in E5.

now, letters L5 and L4 are left and envelopes E1 and E4 are left, so these letters can be put in the incorrect envelope in just one way

so, total combinations are 4*4*3*1*1=48

p=48/120 (which is incorrect).

Am I solving it incorrectly?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9704
Location: Pune, India
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

12 Jun 2017, 09:21
1
3
AR15J wrote:
VeritasPrepKarishma wrote:
vardhanindaram wrote:
5 letters have to be put in 5 different envelopes numbered 1 through 5 such that each of the letters go into only 1 envelope.The letter is said to be put in correct position if for example letter 1 goes into envelope 1.Now what is the probability that all letters be put into wrong envelopes?

A. 1/3
B. 2/3
C. 11/120
D. 11/30
E. 76/120

Note that this question is not a GMAT type question. A GMAT type P&C question would be tricky, (perhaps fun!) but do-able in a minute or two. It would not depend on your knowledge of obscure formulas and would not require you to go through a ton of calculations.
A question which deals with 3 letters and 3 envelopes and asks you to say, find the number of ways in which at least 2 letters go in the correct envelope is far more GMAT friendly.
It would require you to realise that you cannot have exactly 2 letters going in the correct envelope so all you need to do is find the number of ways in which all 3 go in the correct envelope.

This post discusses the logic of putting letters into envelopes: http://www.veritasprep.com/blog/2011/12 ... envelopes/

Hi VeritasPrepKarishma,

Let's say there are 5 letters--L1,L2,L3,L4,L5

and there are 5 envelopes- E1,E2,E3,E4,E5

L1 can be put incorrectly in 4 ways(E2,E3,E4,E5), let's say, it is put in E3
now, L3 can also be put incorrectly in 4 ways(E1,E2,E4,E5), lets say it is put in E2.

Putting the letter L3 in E1 OR in E2/E4/E5 are different cases

If L3 is put in E1, we have 3 letters and 3 envelopes - L2, L4, L5 and E2, E4, E5

All three need to be put in different envelopes. There are 2 ways of doing this (discussed in my blog post)

In this case, there are 4*1*2 = 8 ways of putting all the letters incorrectly.

The rest of your solution goes like this now:

Or L3 can be put in E2/E4/E5 in 3 ways. Let's say it is put in E2.

now, L2 can be put incorrectly in E1, E4 or E5. Again, now if L2 is put in E1, there is only 1 way to put L4 and L5 incorrectly.

But if L2 is put in E4 (or E5), then L4 (or L5) can be put in 2 ways.

So there are in all 3 ways to put L2 and the remaining letters.

so, total combinations are 4*3*3=36

Required Probability = (8 + 36)/120 = 44/120 = 11/30
_________________
Karishma
Veritas Prep GMAT Instructor

Senior Manager
Joined: 21 Aug 2016
Posts: 256
Location: India
GPA: 3.9
WE: Information Technology (Computer Software)
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

12 Jun 2017, 11:21
Awesome explanation !!! Thank you very much
Senior Manager
Joined: 05 Jul 2018
Posts: 424
Location: India
Concentration: General Management, Technology
GMAT 1: 600 Q47 V26
GRE 1: Q162 V149
GPA: 3.6
WE: Information Technology (Consulting)
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

28 Apr 2019, 08:22
A simple derangement method can be used for al such questions where n items have to misplaced in n placeholders
D(n) is the number of ways in which all n items reach wrong place holders. Formula for same is
D(n)= $$N!( 1- 1/1! + 1/2!- 1/3! + 1/4! - ..... +(-1)^n/n!)$$ More on this idea in the video appended at the end
For simplicity one may even remember D(1) till D(6) as 0,1,2,9,44,265

Here total possible ways to put 5 letters in 5 envelopes = 5!= 120
Total favorable derangements for misplacing each letter in wrong envelope= D(5) = 44

Probablity= 44/120= 11/30

_________________
Appreciate any KUDOS given !

MY MBA RESOURCES:

4000 Official Verbal Question | 3700 Official quant questions

Senior Manager
Joined: 13 Feb 2018
Posts: 450
GMAT 1: 640 Q48 V28
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

Show Tags

15 May 2019, 07:51
I'm defeated

Note that this question is not a GMAT type question.
Your words are very welcome. Thank you
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5   [#permalink] 15 May 2019, 07:51
Display posts from previous: Sort by