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:

In how many ways can the letters of the word PERMUTATIONS be [#permalink]

Show Tags

17 May 2010, 14:01

8

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

(N/A)

Question Stats:

64% (01:51) correct
36% (00:52) wrong based on 10 sessions

HideShow timer Statistics

Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

18 May 2010, 01:13

6

This post received KUDOS

Expert's post

10

This post was BOOKMARKED

chandrun wrote:

Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of \(n\) things of which \(P_1\) are alike of one kind, \(P_2\) are alike of second kind, \(P_3\) are alike of third kind ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+..+P_r=n\) is:

\(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\).

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \(\frac{9!}{4!3!2!}\).

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210; 2. Permutation of the letters P ans S (PXXXXS or SXXXXP) = 2! =2; 3. Permutation of the 4 letters between P and S = 4! =24; 4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040; 5. We should divide multiplication of the above 4 numbers by 2! as there is repeated T.

Hence: \(\frac{10C4*2!*4!*7!}{2!}=25,401,600\)

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

There are 11 letters in the word "MISSISSIPPI ", out of which: M=1, I=4, S=4, P=2.

Total # of permutations is \(\frac{11!}{4!4!2!}\); # of permutations with 4 I's together is \(\frac{8!}{4!2!}\). Consider 4 I's as one unit: {M}{S}{S}{S}{S}{P}{P}{IIII} - total 8 units, out of which {M}=1, {S}=4, {P}=2, {IIII}=1.

So # of permutations with 4 I's not come together is: \(\frac{11!}{4!4!2!}-\frac{8!}{4!2!}\).

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

31 Jan 2011, 17:45

1

This post received KUDOS

Expert's post

mariyea wrote:

Why 7!? There are only six letters to arrange...

Are you talking about Q1, 4th point? Can you please be more specific when asking questions?

4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;

There are 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 6 letters in unit 1, plus 6 units with one letter in each = total of 12 letters. _________________

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

01 Feb 2011, 07:07

Bunuel wrote:

mariyea wrote:

Why 7!? There are only six letters to arrange...

Are you talking about Q1, 4th point? Can you please be more specific when asking questions?

4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;

There are 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 6 letters in unit 1, plus 6 units with one letter in each = total of 12 letters.

Yes I am referring to the first q. Sorry about that.

But the q asks for four letters to be placed b/n P and S... and the ways in which the four letters can be arranged is expressed by 4! How can there be six units that have one letter each? _________________

Thank you for your kudoses Everyone!!!

"It always seems impossible until its done." -Nelson Mandela

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

01 Feb 2011, 07:26

2

This post received KUDOS

Expert's post

mariyea wrote:

Yes I am referring to the first q. Sorry about that.

But the q asks for four letters to be placed b/n P and S... and the ways in which the four letters can be arranged is expressed by 4! How can there be six units that have one letter each?

It seems that you don't understand what the question is asking. It does not ask about the ways 4 lettter can be arranged between P and S.

The question is: in how many ways can the word PERMUTATIONS be arranged SO THAT in each arrangement there are always 4 letters between P and S.

This should be calculated in several steps. Step 4 is dealing with arrangement of 7 units: P and S with 4 letters between them (as required) is one unit {PXXXXS} so we used 6 letters, EACH of the rest 6 letters is a separate unit itself so we have total of 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 7!. _________________

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

01 Feb 2011, 07:39

Bunuel wrote:

mariyea wrote:

Yes I am referring to the first q. Sorry about that.

But the q asks for four letters to be placed b/n P and S... and the ways in which the four letters can be arranged is expressed by 4! How can there be six units that have one letter each?

It seems that you don't understand what the question is asking. It does not ask about the ways 4 lettter can be arranged between P and S.

The question is: in how many ways can the word PERMUTATIONS be arranged SO THAT in each arrangement there are always 4 letters between P and S.

This should be calculated in several steps. Step 4 is dealing with arrangement of 7 units: P and S with 4 letters between them (as required) is one unit {PXXXXS} so we used 6 letters, EACH of the rest 6 letters is a separate unit itself so we have total of 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 7!.

Bunuel, Bunuel! Thank you so much! I get it now, you're the man! _________________

Thank you for your kudoses Everyone!!!

"It always seems impossible until its done." -Nelson Mandela

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

03 Feb 2011, 03:42

2

This post received KUDOS

Bunuel wrote:

chandrun wrote:

Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of \(n\) things of which \(P_1\) are alike of one kind, \(P_2\) are alike of second kind, \(P_3\) are alike of third kind ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+..+P_r=n\) is:

\(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\).

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \(\frac{9!}{4!3!2!}\).

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210; 2. Permutation of the letters P ans S (PXXXXS or SXXXXP) = 2! =2; 3. Permutation of the 4 letters between P and S = 4! =24; 4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040; 5. We should divide multiplication of the above 4 numbers by 2! as there is repeated T.

Hence: \(\frac{10C4*2!*4!*7!}{2!}=25,401,600\)

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

There are 11 letters in the word "MISSISSIPPI ", out of which: M=1, I=4, S=4, P=2.

Total # of permutations is \(\frac{11!}{4!4!2!}\); # of permutations with 4 I's together is \(\frac{8!}{4!2!}\). Consider 4 I's as one unit: {M}{S}{S}{S}{S}{P}{P}{IIII} - total 8 units, out of which {M}=1, {S}=4, {P}=2, {IIII}=1.

So # of permutations with 4 I's not come together is: \(\frac{11!}{4!4!2!}-\frac{8!}{4!2!}\).

Hope it helps.

What about my approach to PERMUTATIONS?

The word permutations consists of 12 letters. You can choose P and S in 7*2 ways so there are always 4 numbers between them P on first, S on fifth ........ P on seventh S on twelwth + reversely (SP)

You are left with 10 letters, 2 of which are the same (TT)

So the complete formula is:

\(7*2*\frac{10!}{2!}=7*10!\)

It makes the same result, but I think is a bit quicker. _________________

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

03 Feb 2011, 03:48

Expert's post

craky wrote:

Bunuel wrote:

chandrun wrote:

Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of \(n\) things of which \(P_1\) are alike of one kind, \(P_2\) are alike of second kind, \(P_3\) are alike of third kind ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+..+P_r=n\) is:

\(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\).

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \(\frac{9!}{4!3!2!}\).

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210; 2. Permutation of the letters P ans S (PXXXXS or SXXXXP) = 2! =2; 3. Permutation of the 4 letters between P and S = 4! =24; 4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040; 5. We should divide multiplication of the above 4 numbers by 2! as there is repeated T.

Hence: \(\frac{10C4*2!*4!*7!}{2!}=25,401,600\)

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

There are 11 letters in the word "MISSISSIPPI ", out of which: M=1, I=4, S=4, P=2.

Total # of permutations is \(\frac{11!}{4!4!2!}\); # of permutations with 4 I's together is \(\frac{8!}{4!2!}\). Consider 4 I's as one unit: {M}{S}{S}{S}{S}{P}{P}{IIII} - total 8 units, out of which {M}=1, {S}=4, {P}=2, {IIII}=1.

So # of permutations with 4 I's not come together is: \(\frac{11!}{4!4!2!}-\frac{8!}{4!2!}\).

Hope it helps.

What about my approach to PERMUTATIONS?

The word permutations consists of 12 letters. You can choose P and S in 7*2 ways so there are always 4 numbers between them P on first, S on fifth ........ P on seventh S on twelwth + reversely (SP)

You are left with 10 letters, 2 of which are the same (TT)

So the complete formula is:

\(7*2*\frac{10!}{2!}=7*10!\)

It makes the same result, but I think is a bit quicker.

Re: Interesting problems of Permutations and Combinations [#permalink]

Show Tags

04 Feb 2011, 06:41

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

PERMUTATIONS: Total length of the word = 12 Repetitions: T:2. Rest all letters:1

P,S can occupy following indices in the word {1,6},{2,7},{3,8},{4,9},{5,10},{6,11},{7,12}

In each of the above positions that P and S occupy, the remaining 10 letters can be arranged in \(\frac{10!}{2!}\) ways

Likewise; S,P can occupy following indices: {1,6},{2,7},{3,8},{4,9},{5,10},{6,11},{7,12}

In each of the above positions that S and P occupy, the remaining 10 letters can be arranged in \(\frac{10!}{2!}\) ways

So, total number of arrangements:

\(\frac{7*2*10!}{2!}\)

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Word MISSISSIPPI Length: 11 Repetitions: I:4 S:4 P:2 M:1

Total number of arrangement possible = \(\frac{11!}{4!4!2!}\) ------------Ist

Now, let's conjoin four I's and treat it as a unique character, say #. We have conjoined all fours in order to symbolize that all I's are adhered together.

So, Now MISSISSIPPI becomes M#SSSSPP Word: M#SSSSPP Length: 8 Repetitions: S:4 P:2 M:1 #:1

The number of ways in which I's come together is: \(\frac{8!}{2!4!}\) --------- 2nd

Thus, the number of arrangements in which I's not come together will be the difference between 2nd and Ist

Re: In how many ways can the letters of the word PERMUTATIONS be [#permalink]

Show Tags

17 Jul 2014, 22:08

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: In how many ways can the letters of the word PERMUTATIONS be [#permalink]

Show Tags

28 Aug 2014, 22:19

In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S? I solved this way: P_ _ _ _ S _ _ _ _ _ _

10! [ arrange remaining 10 letters] divided by 2! [ because of 2 T] multiplied by 2! [ swap P and S] multiplied by 7 [ P and S can move 6 places ahead]

Re: In how many ways can the letters of the word PERMUTATIONS be [#permalink]

Show Tags

28 Dec 2015, 01:35

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: In how many ways can the letters of the word PERMUTATIONS be [#permalink]

Show Tags

05 Jun 2016, 03:13

Bunuel wrote:

chandrun wrote:

Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of \(n\) things of which \(P_1\) are alike of one kind, \(P_2\) are alike of second kind, \(P_3\) are alike of third kind ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+..+P_r=n\) is:

\(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\).

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \(\frac{9!}{4!3!2!}\).

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210;

Hi,

As you have chosen 4 letters out of 10 letters which include two TT's, would it not lead to duplication of the selections? As in the case of "ENTRANCE", if we had to choose the number of selections, we would go as follows -

We have 8 letters from which 6 are unique.

Possible scenarios for 4 letter selection are: A. All letters are different; B. 2 N-s and other letters are different; C. 2 E-s and other letters are different; D. 2 N-s, 2 E-s.

Let's count them separately: A. All letters are different, means that basically we are choosing 4 letters form 6 distinct letters: 6C4=15; B. 2 N-s and other letters are different: 2C2(2 N-s out of 2)*5C2(other 2 letters from distinct 5 letters left)=10; C. 2 E-s and other letters are different: 2C2(2 E-s out of 2)*5C2(other 2 letters from distinct 5 letters left)=10;; D. 2 N-s, 2 E-s: 2C2*2C2=1.

15+10+10+1=36

gmatclubot

Re: In how many ways can the letters of the word PERMUTATIONS be
[#permalink]
05 Jun 2016, 03:13

Excellent posts dLo saw your blog too..!! Man .. you have got some writing skills. And Just to make an argument = You had such an amazing resume ; i am glad...

So Much $$$ Business school costs a lot. This is obvious, whether you are a full-ride scholarship student or are paying fully out-of-pocket. Aside from the (constantly rising)...

I barely remember taking decent rest in the last 60 hours. It’s been relentless with submissions, birthday celebration, exams, vacating the flat, meeting people before leaving and of...

Rishabh from Gyan one services, India had a one to one interview with me where I shared my experience at IMD till now. http://www.gyanone.com/blog/life-at-imd-interview-with-imd-mba/ ...