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# In how many different ways can the letters of the word 'READING' be a

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In how many different ways can the letters of the word 'READING' be a  [#permalink]

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20 Mar 2017, 06:11
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Q. In how many different ways can the letters of the word 'READING' be arranged to form new words such that A comes before E and E before I?

A. 120
B. 840
C. 1680
D. 2520
E. 5040

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Saquib
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In how many different ways can the letters of the word 'READING' be a  [#permalink]

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18 Oct 2018, 19:02
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7!/3! Is one of the easiest ways to calculate. Or else, if anyone need to understand verbally then total no of ways of selecting three positions out of 7 is 7C3. This includes AEI together and AEI with in between letters.

So now remaining 4 letters can be arranged in 4! Ways.

So 7C3 x 4! = 840

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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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20 Mar 2017, 06:12
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In how many different ways can the letters of the word 'READING' be a  [#permalink]

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20 Mar 2017, 09:43
IMO Ans = 120.
READING = 7 LETTERS, if we remove AEI, there are only 4 letters remaining.
1 AEI _ _ _ _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
2 _ AEI _ _ _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
3. _ _ AEI _ _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
4. _ _ _ AEI _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
5. _ _ _ _AEI (No . of ways of arranging 4 letters in 4 spaces = 4p4)

Total no. ways of doing the required task = 5 X4p4 = 5 X 4! = 5! = 120.

Please let me know whether i am correct or not.

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Narayana Raju
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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20 Mar 2017, 15:30
A

We consider AEI 1 unit because they will always be together
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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20 Mar 2017, 22:41
gvvsnraju@1 wrote:
IMO Ans = 120.
READING = 7 LETTERS, if we remove AEI, there are only 4 letters remaining.
1 AEI _ _ _ _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
2 _ AEI _ _ _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
3. _ _ AEI _ _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
4. _ _ _ AEI _ (No . of ways of arranging 4 letters in 4 spaces = 4p4)
5. _ _ _ _AEI (No . of ways of arranging 4 letters in 4 spaces = 4p4)

Total no. ways of doing the required task = 5 X4p4 = 5 X 4! = 5! = 120.

Please let me know whether i am correct or not.

Regards
Narayana Raju

Hi Raju ,

You found the ways while considering no other Letters are in between AEI, however question is just asking to find out the ways such that letter A comes before E and E before I
So resulting words can be formed such as
ARENIDG ARGENID RAEINGD (few examples)

That's why the QA - B (840 ways calculated as 7!/3! ) considering all possible situations in which A comes before E and E comes before I.

Hope this makes question more clear.

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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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21 Mar 2017, 02:42
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EgmatQuantExpert wrote:
Q. In how many different ways can the letters of the word 'READING' be arranged to form new words such that A comes before E and E before I?

A. 120
B. 840
C. 1680
D. 2520
E. 5040

Thanks,
Saquib
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no. of ways in which 7 letters can be arranged = 7!
no. of ways in which those 3 letters can be arranged = 3!

since these 3 letters will always have only one arrange ment
therefore total ways = 7!/3! = 840

Option B
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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16 Oct 2018, 01:56
why did they divide by 3 factorial ??
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In how many different ways can the letters of the word 'READING' be a  [#permalink]

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16 Oct 2018, 06:53
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siddharthfrancis wrote:
why did they divide by 3 factorial ??

No of ways to arrange all the letters in READING = 7!
This contains all the permutation in which A < E < I or A < I < E or I < E < A etc.

No of ways to arrange A , E , I = 3! .
Only in one of these A will come before E & I and E before I.

Therefore 7!/3!
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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18 Oct 2018, 18:31
Could someone please explain why we are dividing by 3!?
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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19 Jan 2019, 10:47
My understanding is as below:

we can select 3 places from the given 7 in 7C3 ways.Now,in the selection,there in only 1 way to arrange A,E,I in the desired sequence.Remaining 4 letters can be arranged in 4! ways.

So answer should be: 7C3* 4!= 7!/3!=840

EgmatQuantExpert chetan2u am I correct in my reasoning?
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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19 Jan 2019, 11:39
1
arpitkansal wrote:
My understanding is as below:

we can select 3 places from the given 7 in 7C3 ways.Now,in the selection,there in only 1 way to arrange A,E,I in the desired sequence.Remaining 4 letters can be arranged in 4! ways.

So answer should be: 7C3* 4!= 7!/3!=840

EgmatQuantExpert chetan2u am I correct in my reasoning?

Yes, you are correct.

Another way to look at it is that the 7 words can be arranged in 7! Ways..
But 3 of them can be arranged in just 1 way rather than 3! Ways.
Therefore 7!/3! Ways
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Re: In how many different ways can the letters of the word 'READING' be a  [#permalink]

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20 Jan 2019, 01:34
READING ; 7 letters which can be arranged in 7! ways
we need to find ways READING can be written where sequence of AEI is there
AEIRDNG ; eg so we can put AEI as X
XRDNG ; it can be written in 5! ways and placement of X can be in following order

RXDNG
RDXNG
RDNXG
RDNGX

5!*4 : 120*4 ; 840
IMO B

EgmatQuantExpert wrote:
Q. In how many different ways can the letters of the word 'READING' be arranged to form new words such that A comes before E and E before I?

A. 120
B. 840
C. 1680
D. 2520
E. 5040

Thanks,
Saquib
Quant Expert
e-GMAT

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Re: In how many different ways can the letters of the word 'READING' be a   [#permalink] 20 Jan 2019, 01:34
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