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

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5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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Updated on: 10 Mar 2016, 00:23
8
00:00

Difficulty:

95% (hard)

Question Stats:

27% (01:50) correct 73% (02:50) wrong based on 79 sessions

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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.
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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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

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5-letters-have-to-be-put-into-their-5-respective-envelopes-189501.html
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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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

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Posts: 6961
Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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10 Mar 2016, 01:15
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
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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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/
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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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?
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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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
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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12 Jun 2017, 11:21
Awesome explanation !!! Thank you very much
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Re: 5 letters have to be put in 5 different envelopes numbered 1 through 5  [#permalink]

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30 Sep 2018, 06:55
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