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# Letters of the word “POLITICS” are shuffled to form all possible

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Senior CR Moderator
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Joined: 10 Oct 2016
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Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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15 Jul 2017, 08:50
18
00:00

Difficulty:

95% (hard)

Question Stats:

32% (02:11) correct 68% (01:58) wrong based on 82 sessions

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Letters of the word “POLITICS” are shuffled to form all possible distinct combinations. In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

A. 360
B. 720
C. 2,520
D. 3,360
E. 6,720

Source: ExpertsGlobal

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Math Expert
Joined: 02 Aug 2009
Posts: 6565
Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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15 Jul 2017, 09:10
2
4
broall wrote:
Letters of the word “POLITICS” are shuffled to form all possible distinct combinations. In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

A. 360
B. 720
C. 2,520
D. 3,360
E. 6,720

Source: ExpertsGlobal

Hi,

POLITICS can be arranged in $$\frac{8!}{2!}$$..
Now P,O,L can be arranged within themselves in 3! But ONLY one will have P before O before L..
So ANSWER is $$\frac{8!}{2*3!}=3360$$
D
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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##### General Discussion
Intern
Joined: 13 Jul 2017
Posts: 10
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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15 Jul 2017, 15:29
This is how I got to my solution:

P - - - - - - -

Considering the letter P on the 1st slot we have a total of:

5!*6 + 5!*5 + 5!*4 + 5!*3 + 5!*2 + 5!*1 = 2520 possibilities if O is after P and L is after O

For P in the second slot:

- P - - - - - -

5!*5 + 5!*4 + 5!*3 + 5!*2 + 5!*1 = 1800

We can do this until P is on the 3rd to last slot so that we have room for O and L after it.

5!*(6+5+4+3+2+1) + 5!*(5+4+3+2+1) + 5!*(4+3+2+1) + 5!*(3+2+1) + 5!*(2+1) + 5!*(1) = 6720

Im sure there is an easier way to do what I did but my biggest problem is: Why is it wrong?
Math Expert
Joined: 02 Aug 2009
Posts: 6565
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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15 Jul 2017, 21:16
karek77 wrote:
This is how I got to my solution:

P - - - - - - -

Considering the letter P on the 1st slot we have a total of:

5!*6 + 5!*5 + 5!*4 + 5!*3 + 5!*2 + 5!*1 = 2520 possibilities if O is after P and L is after O

For P in the second slot:

- P - - - - - -

5!*5 + 5!*4 + 5!*3 + 5!*2 + 5!*1 = 1800

We can do this until P is on the 3rd to last slot so that we have room for O and L after it.

5!*(6+5+4+3+2+1) + 5!*(5+4+3+2+1) + 5!*(4+3+2+1) + 5!*(3+2+1) + 5!*(2+1) + 5!*(1) = 6720

Im sure there is an easier way to do what I did but my biggest problem is: Why is it wrong?

Hi,

the main place where you have gone wrong is that you have NOT catered for two Is...
If a set of digits have two similar digit or letter, you are required to divide solution by 2 for removing the redundancy.
so 6720/2=3360

hope it clears the doubt...
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Intern
Joined: 13 Jul 2017
Posts: 10
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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15 Jul 2017, 21:22
chetan2u wrote:
karek77 wrote:
This is how I got to my solution:

P - - - - - - -

Considering the letter P on the 1st slot we have a total of:

5!*6 + 5!*5 + 5!*4 + 5!*3 + 5!*2 + 5!*1 = 2520 possibilities if O is after P and L is after O

For P in the second slot:

- P - - - - - -

5!*5 + 5!*4 + 5!*3 + 5!*2 + 5!*1 = 1800

We can do this until P is on the 3rd to last slot so that we have room for O and L after it.

5!*(6+5+4+3+2+1) + 5!*(5+4+3+2+1) + 5!*(4+3+2+1) + 5!*(3+2+1) + 5!*(2+1) + 5!*(1) = 6720

Im sure there is an easier way to do what I did but my biggest problem is: Why is it wrong?

Hi,

the main place where you have gone wrong is that you have NOT catered for two Is...
If a set of digits have two similar digit or letter, you are required to divide solution by 2 for removing the redundancy.
so 6720/2=3360

hope it clears the doubt...

Errr cant believe I didnt see this. Thank you
Intern
Joined: 28 Apr 2017
Posts: 42
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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16 Jul 2017, 01:18
nguyendinhtuong wrote:
Letters of the word “POLITICS” are shuffled to form all possible distinct combinations. In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

A. 360
B. 720
C. 2,520
D. 3,360
E. 6,720

Source: ExpertsGlobal

Sol: Other way to look at this problem is to take P,O,L and arrange them =8C3*1...(1 because there will be only 1 arrangement when they will be in required sequence).
And then filling the left out positions=5!/2!....5 Letters and divided by 2 for 2 I's.
=8C3*1*(5!/2!)=8!/2!3!
Intern
Joined: 10 Jun 2018
Posts: 6
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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21 Jul 2018, 04:16
I can't seem to understand what is wrong with this approach. I am really bad at P&C, so maybe my concept is altogether wrong. It would be great if someone can explain it to me.
So I tied P,O,L together as a string which can be arranged only in 1 way.
Now we have to arrange POL,I,T,I,C,S which can be done in 6! ways and as 'I' is present 2 times, we divide by 2!
So the answer I reach to is 6!/2!= 360
Manager
Joined: 17 May 2015
Posts: 237
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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21 Jul 2018, 05:12
aanchalk wrote:
I can't seem to understand what is wrong with this approach. I am really bad at P&C, so maybe my concept is altogether wrong. It would be great if someone can explain it to me.
So I tied P,O,L together as a string which can be arranged only in 1 way.
Now we have to arrange POL,I,T,I,C,S which can be done in 6! ways and as 'I' is present 2 times, we divide by 2!
So the answer I reach to is 6!/2!= 360

Hi aanchalk,

The question asks In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

It didn't say that P, O, and L should be together. In the above solution, many cases are not counted. For example:

P I O T L I C S

I hope this helps.

Thanks.
Intern
Joined: 10 Jun 2018
Posts: 6
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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21 Jul 2018, 05:52
ganand wrote:
aanchalk wrote:
I can't seem to understand what is wrong with this approach. I am really bad at P&C, so maybe my concept is altogether wrong. It would be great if someone can explain it to me.
So I tied P,O,L together as a string which can be arranged only in 1 way.
Now we have to arrange POL,I,T,I,C,S which can be done in 6! ways and as 'I' is present 2 times, we divide by 2!
So the answer I reach to is 6!/2!= 360

Hi aanchalk,

The question asks In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

It didn't say that P, O, and L should be together. In the above solution, many cases are not counted. For example:

P I O T L I C S

I hope this helps.

Thanks.

Oh I got it now! Thanks a ton for clearing that up! I confused myself after a point
Manager
Joined: 02 Jul 2016
Posts: 118
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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29 Jul 2018, 03:47
chetan2u wrote:
broall wrote:
Letters of the word “POLITICS” are shuffled to form all possible distinct combinations. In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

A. 360
B. 720
C. 2,520
D. 3,360
E. 6,720

Source: ExpertsGlobal

Hi,

POLITICS can be arranged in $$\frac{8!}{2!}$$..
Now P,O,L can be arranged within themselves in 3! But ONLY one will have P before O before L..
So ANSWER is $$\frac{8!}{2*3!}=3360$$
D

Hi chetan2u
Thanks for the explanation,
Can you please elaborate the highlighted part???

Math Expert
Joined: 02 Aug 2009
Posts: 6565
Re: Letters of the word “POLITICS” are shuffled to form all possible  [#permalink]

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29 Jul 2018, 04:00
suramya26 wrote:
chetan2u wrote:
broall wrote:
Letters of the word “POLITICS” are shuffled to form all possible distinct combinations. In how many cases does ‘P’ come before ‘O’, and ‘O’ come before ‘L’?

A. 360
B. 720
C. 2,520
D. 3,360
E. 6,720

Source: ExpertsGlobal

Hi,

POLITICS can be arranged in $$\frac{8!}{2!}$$..
Now P,O,L can be arranged within themselves in 3! But ONLY one will have P before O before L..
So ANSWER is $$\frac{8!}{2*3!}=3360$$
D

Hi chetan2u
Thanks for the explanation,
Can you please elaborate the highlighted part???

Anything having 8 different characters can be arranged in 8! Ways..
But if two of the eight characters are same then it can be arranged in 8!/2! as these 2 can be arranged in 2! Ways and in 8! each has been taken separately.

Now take any one of these say POLITICS itself..
Freeze all other except P,O,L
So _,_,_,I,T,I,C,S
The three blank have to be filled by P,O,L and these three can be arranged in 3! ways but only one will have P before O and O before L so in 8!/2!, one pattern has been written in 3! ways, so 8!/(2!3!)

Take another way ... STIIOLCP.. freeze all except POL so STII_,_C,_
Again these three blanks can be filled in 3! ways but only one will be having P,O and L in this order and that is STIIPOCL
Therefore we divide it by 3! And get 8!/(2!3!)
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Re: Letters of the word “POLITICS” are shuffled to form all possible &nbs [#permalink] 29 Jul 2018, 04:00
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