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

Question

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

chetan2u · Accepted Answer

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

karek77 · Answer

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.

So my answer would be:

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

Answer E

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

chetan2u · Answer

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...

karek77 · Answer

Errr cant believe I didnt see this. Thank you

deepudiscover · Answer

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!

aanchalk · Answer

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
Please help me with this.

ganand · Answer

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.

aanchalk · Answer

Oh I got it now! Thanks a ton for clearing that up! I confused myself after a point

suramya26 · Answer

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

Thanks in advance

chetan2u · Answer

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!)

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards