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 Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

I myself do not know whether I am right. Your answer seems to be too large, close to 1. But we deal with an exceptional case--the probability should be small.

I base my solution on the assumption that each letter has 5 chances to take a wrong envelope and the only chance to take a right one.

BTW, the probability that all the letters will be placed correctly is 1/6!

There are 6 letters and 6 self addressed envelopes.What is the probability that none is correctly placed.

I believe the answer is 265/720 = 53/144.
_________________

Best,

AkamaiBrah Former Senior Instructor, Manhattan GMAT and VeritasPrep Vice President, Midtown NYC Investment Bank, Structured Finance IT MFE, Haas School of Business, UC Berkeley, Class of 2005 MBA, Anderson School of Management, UCLA, Class of 1993

Last edited by AkamaiBrah on 30 Jul 2003, 05:47, edited 1 time in total.

There are 6 letters and 6 self addressed envelopes.What is the probability that none is correctly placed.

Consider this:

I believe the answer is 265/720 = 53/144.

Akamai,

Please elaborate on your answer

Solution:

First of all, IMO, although this problem looks basic, in reality this problem is WAY WAY WAY too difficult for the GMAT and will stump even graduate math students (it took me about 30 minutes).

We are looking for the solution of G(6) where G(n) = number of ways n items can be permuted such that they none of them match their original position. This is the same as having a deck of n cards numbered from 1 to n, then counting the ways you can shuffle the deck without any cards matching its position.

Consider this: it is obvious that when n = 1, G(n) must be ZERO, and when n = 2, G(n) = 1 (i.e., {2,1}), and when n = 3, G(n) = 2 (i.e., {2,3,1},{3,1,2}). By drawing out a probability tree, you will discover that G(4) = 9, and with a LOT of work and patience, G(5) = 44.

If you carefully watch how the tree is growing, you may notice that the results of G(n) refer back to the results of G(n-1) and G(n-2). With a little bit of work, we can see that a general recursive formula for G(n) is:

G(n) = (n-1)*(G(n-1)+G(n-2)) where n > 1 and G(1) = 0 and G(2) = 1.

If you build a table, then G(6) comes out to 5*(44 + 9) = 265. Since there are 6! or 720 ways to stuff the envelopes, the probabiltiy of getting them all wrong is 265/720 = 53/144.

(if you don't believe me, run a simulation).
_________________

Best,

AkamaiBrah Former Senior Instructor, Manhattan GMAT and VeritasPrep Vice President, Midtown NYC Investment Bank, Structured Finance IT MFE, Haas School of Business, UC Berkeley, Class of 2005 MBA, Anderson School of Management, UCLA, Class of 1993

Last edited by AkamaiBrah on 30 Jul 2003, 06:12, edited 1 time in total.

There are 6 letters and 6 self addressed envelopes.What is the probability that none is correctly placed.

Consider this:

I believe the answer is 265/720 = 53/144.

Akamai,

Please elaborate on your answer

Solution:

First of all, IMO, although this problem looks basic, in reality this problem is WAY WAY WAY too difficult for the GMAT and will stump even graduate math students (it took me about 30 minutes).

We are looking for the solution of G(6) where G(n) = number of ways n items can be permuted such that they none of them match their original position. This is the same as having a deck of n cards numbered from 1 to n, then counting the ways you can shuffle the deck without any cards matching its position.

Consider this: it is obvious that when n = 1, G(n) must be ZERO, and when n = 2, G(n) = 1 (i.e., {2,1}), and when n = 3, G(n) = 2 (i.e., {2,3,1},{3,1,2}). By drawing out a probability tree, you will discover that G(4) = 9, and with a LOT of work and patience, G(5) = 44.

If you carefully watch how the tree is growing, you may notice that the results of G(n) refer back to the results of G(n-1) and G(n-2). With a little bit of work, we can see that a general recursive formula for G(n) is:

G(n) = (n-1)*(G(n-1)+G(n-2)) where n > 1 and G(1) = 0 and G(2) = 1.

If you build a table, then G(6) comes out to 5*(44 + 9) = 265. Since there are 6! or 720 ways to stuff the envelopes, the probabiltiy of getting them all wrong is 265/720 = 53/144.

(if you don't believe me, run a simulation).

One of the most interesting problem i have come across!!!

I saw a similar question: An inefficient secretary places n different letters into n differently addressed envelopes at random.Find the probability that atleast one of the letters will arrive at the proper destination.
P=C(n,1)1/n-C(n,2)1/n*1/n-1 .......(-1)^n-1*C(n,n)*1/n!
=1-1/2!+1/3!..... +(-1)^n-1*1/n!

I saw a similar question: An inefficient secretary places n different letters into n differently addressed envelopes at random.Find the probability that atleast one of the letters will arrive at the proper destination. P=C(n,1)1/n-C(n,2)1/n*1/n-1 .......(-1)^n-1*C(n,n)*1/n! =1-1/2!+1/3!..... +(-1)^n-1*1/n!

Sanjay

I don't have time to derive the result you got, but if you plug in the numbers, it is consistent with my solution for any n. (The solution for this problem is 1 minus the probability for the original problem).

While this is an interesting problem, it is not a GMAT problem, nor does it yield any insights to help with GMAT problems, so I will leave it at this.
_________________

Best,

AkamaiBrah Former Senior Instructor, Manhattan GMAT and VeritasPrep Vice President, Midtown NYC Investment Bank, Structured Finance IT MFE, Haas School of Business, UC Berkeley, Class of 2005 MBA, Anderson School of Management, UCLA, Class of 1993

Re: There are 6 letters and 6 self addressed envelopes.What is [#permalink]

Show Tags

29 Oct 2012, 03:38

HI Prashant, You are getting 637/720 since you are counting repeated combinations. Here goes the solution 6 cards need to be placed in 6 envelops such that not even one card is in the right envelop. Let us assume that atleast one card is in the right envelop. Number of ways in which atleast one card is placed in the right envelop is 6C1 * (6-1)! ways. [We select one card and place it in its right envelop and arrange the rest of the 5 cards in the remaining 5 envelops.] This should be subtracted from 6! However in the above cases where atleast one card is placed in the right envelop, the cases where exactly two cards are placed in the right envelop are counted twice. Since we cannot find the number of cases where exactly two cards are placed in the correct envelops, we add back the cases where atleast 2 cards are placed in the right envelops which is 6C2 (6-2)! While adding the above cases, all the cases where exactly 3 cards are placed in the right envelops are also added back which should be subtracted. Thus we see a pattern of subtractions and additions . Thus the number of ways is 6!-6C1(5!)+6C2(4!)-6C3(3!)+6C4(2!)+6C5(1!)+6C6(0!)= 256 ways. Thus the probability that none of them are in the correct envelops is 256/720 ways.

gmatclubot

Re: There are 6 letters and 6 self addressed envelopes.What is
[#permalink]
29 Oct 2012, 03:38

Happy New Year everyone! Before I get started on this post, and well, restarted on this blog in general, I wanted to mention something. For the past several months...

It’s quickly approaching two years since I last wrote anything on this blog. A lot has happened since then. When I last posted, I had just gotten back from...

Happy 2017! Here is another update, 7 months later. With this pace I might add only one more post before the end of the GSB! However, I promised that...

The words of John O’Donohue ring in my head every time I reflect on the transformative, euphoric, life-changing, demanding, emotional, and great year that 2016 was! The fourth to...