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

 It is currently 12 Dec 2018, 04:02

### 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

## Events & Promotions

###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### The winning strategy for 700+ on the GMAT

December 13, 2018

December 13, 2018

08:00 AM PST

09:00 AM PST

What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
• ### GMATbuster's Weekly GMAT Quant Quiz, Tomorrow, Saturday at 9 AM PST

December 14, 2018

December 14, 2018

09:00 AM PST

10:00 AM PST

10 Questions will be posted on the forum and we will post a reply in this Topic with a link to each question. There are prizes for the winners.

# A secretary types 4 letters and then addresses the 4

Author Message
Manager
Joined: 09 Jul 2007
Posts: 213

### Show Tags

25 Aug 2008, 18:07
00:00

Difficulty:

(N/A)

Question Stats:

100% (00:00) correct 0% (00:00) wrong based on 14 sessions

### HideShow timer Statistics

A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope?

A) 8
B) 9
C) 10
D) 12
E) 15

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Intern
Joined: 25 Aug 2008
Posts: 3

### Show Tags

25 Aug 2008, 18:42
the ans is 12 ie D

the correct arrangement of letter is 1 1 1 1
wrongly placed is it can be done by3 3 3 3 ways so 4*3 = 12 ways
Manager
Joined: 14 Jun 2008
Posts: 159

### Show Tags

25 Aug 2008, 23:12
vtselvan wrote:
the ans is 12 ie D

the correct arrangement of letter is 1 1 1 1
wrongly placed is it can be done by3 3 3 3 ways so 4*3 = 12 ways

could you expand on that?
what do u mean by wrongly placed is it can be done by3 3 3 3 ?
Senior Manager
Joined: 06 Apr 2008
Posts: 382

### Show Tags

25 Aug 2008, 23:54
ssandeepan wrote:
A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope?

A) 8
B) 9
C) 10
D) 12
E) 15

Total number of ways = 3 + 3 + 3 + 3 = 12
Manager
Joined: 09 Jul 2007
Posts: 213

### Show Tags

26 Aug 2008, 02:51
I was checking if anybody has an easier explanation . Otherwise it goes like this :

answer = total number of ways of arranging letters - (one letter in right envelope + 2 letters in right envelope+ 3 letters in right envelope + 4 letters in right envelope)

Now,
Let the no. of ways in which exactly 1 letter goes into the right envelope & rest of the 3 letters go into 3 wrong envelopes be x1
Let the no. of ways in which exactly 2 letters go into the right envelopes & rest of the 2 letters go into 2 wrong envelopes be x2
Let the no. of ways in which exactly 3 letters go into the right envelopes & 1 letter going into wrong envelope IS NOT POSSIBLE
Let the no. of ways in which all 4 letters go into right envelopes be x4 (only 1 way though)

So, the no. of ways in which all 4 letters go into wrong envelopes = 4! - (x1 + x2 + x4)

Calculating x1 itself is another problem by its own!
i.e, x1 = 4c1 * <No. of ways in which rest of the 3 letters go into 3 wrong envelopes>

And for 3 letters to go into 3 wrong envelopes, we again have 3! - (1 right letter but 2 wrong letters + all 3 right letters) i.e, 3! - (3+1) i.e., 2 ways

So, x1 = 4c1*2 = 8

Coming to x2; 2 letters going into right envelopes can be chosen by 4c2 ways and the rest 2 going wrong is only 1 possibility (interchanging other two)

So, x2 = 4c2*1 = 6

And we know x3 = 1

Hence, the no. of ways in which all 4 letters go into wrong envelopes = 4! - (8+6+1) = 4!-15 = 9 ways
GMAT Instructor
Joined: 04 Jul 2006
Posts: 1251

### Show Tags

26 Aug 2008, 03:03
ssandeepan wrote:
A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope?

A) 8
B) 9
C) 10
D) 12
E) 15

I get only 9:

Envelope A can get b,c or d:
If A gets b, B can get a,c, or d. If B gets a, then c goes in D and d in C. If B gets c, then d goes in C and a goes in D
If A gets c, B can get a or d. If B gets a, then C gets d and D gets b. If B gets d, then C and D get a and b or vice versa.
If A gets d, B can get a or c. If B gets a, then C gets b and D gets C. If B gets c, then C and D get a and b or vice versa.

Perhaps it is better to count the converse: 1 way for 4 correct, 0 ways for 3 correct, 4C2 = 6 ways for 2 correct and 8 ways for only one correct ( 4C1 multiplied by 2). 24 - 1 - 6 - 8 = 9

I like this:
Think of two sets: one with a and b, the other with c and d

3 cases:

case 1: no exchange of letters between the sets- the two sets must each do the only inward swap possible (1)
case 2: one exchange of letters between the sets 2C1 x 2C1 (choose one letter for each set for the exchange, after the exchange, two sets must each do the inward swap (4)
case 3: each letter in the first set moves to the second and vice versa- the new sets can be permuted in 2 ways each (4)
4 + 4 + 1 = 9
Current Student
Joined: 11 May 2008
Posts: 545

### Show Tags

26 Aug 2008, 10:18
1
E1 E2 E3 E4- envelopes
LI L2 L3 L4 - letter

total possible ways = 4! = 24

preffered ways
(eg, when we have L1 in E1)
E1 E2 E3 E4
L1 L4 L2 L3
L1 L2 L4 L3
L1 L3 L2 L4
SO for 1 position of L1, we can have correspoding 3 ways which we dont want the letter arrangement. since there are 4 letter,
4 *3= 12 ways which we dont want.
so no of ways which we want = total poss ways - total unwanted ways = 24-12 = 12

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Non-Human User
Joined: 09 Sep 2013
Posts: 9128
Re: A secretary types 4 letters and then addresses the 4  [#permalink]

### Show Tags

18 Aug 2018, 20:46
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: A secretary types 4 letters and then addresses the 4 &nbs [#permalink] 18 Aug 2018, 20:46
Display posts from previous: Sort by