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# In how many ways can you arrange the letters "APTITUDE" so as to form

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Math Expert
Joined: 02 Sep 2009
Posts: 64082
In how many ways can you arrange the letters "APTITUDE" so as to form  [#permalink]

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17 Sep 2017, 02:56
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Difficulty:

75% (hard)

Question Stats:

59% (01:58) correct 41% (02:05) wrong based on 111 sessions

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In how many ways can you arrange the letters "APTITUDE" so as to form words that begin and end with a vowel?

A. 640
B. 1280
C. 4320
D. 8640
E. 9860

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Math Expert
Joined: 02 Aug 2009
Posts: 8586
Re: In how many ways can you arrange the letters "APTITUDE" so as to form  [#permalink]

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17 Sep 2017, 03:25
Bunuel wrote:
In how many ways can you arrange the letters "APTITUDE" so as to form words that begin and end with a vowel?

A. 640
B. 1280
C. 4320
D. 8640
E. 9860

Hi...

The first letter of the word can be ANY of the 4 vowels and the last letter could be any of the remaining 3..
So first and last letter can be selected in 4*3 ways.
So ans has to be a MULTIPLE of 3... Only C and D are left ( just for info)

Now remaining 6 can be selected in 6!/2!=360... Div by 2 is because of 2 I..

Ans 360*12=4320

C
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form  [#permalink]

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17 Sep 2017, 03:28
Bunuel wrote:
In how many ways can you arrange the letters "APTITUDE" so as to form words that begin and end with a vowel?

A. 640
B. 1280
C. 4320
D. 8640
E. 9860

We have 4 vowels, as order matters in this case # of arrangements of 4 vowels in 2 slots (begin/end) will be 4P2=12

Now we are left with 8-2=6 letters, from which 2 letters are the same (T). # of arrangements 6!/2!=360

Total=360*12=4320

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Re: In how many ways can you arrange the letters "APTITUDE" so as to form  [#permalink]

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18 Sep 2017, 03:08
1
Bunuel wrote:
In how many ways can you arrange the letters "APTITUDE" so as to form words that begin and end with a vowel?

A. 640
B. 1280
C. 4320
D. 8640
E. 9860

There are 4 vowels AIEU
SO 1st letter can be taken in 4 ways
Last letter can be taken in 3 ways
Middle letters can be arranged in 6! ways from 6 letters left.

So total ways = 4*3*6!

But there are 2 T's so we need to divide total arrangement by 2! to compensate the duplicate of words.

Total required no. words formed = 4*3*6!/2! = 4320

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Re: In how many ways can you arrange the letters "APTITUDE" so as to form  [#permalink]

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22 Sep 2017, 06:28
Bunuel wrote:
In how many ways can you arrange the letters "APTITUDE" so as to form words that begin and end with a vowel?

A. 640
B. 1280
C. 4320
D. 8640
E. 9860

If the first letter has to be a vowel, then it can be any one of the 4 vowels (A, E, I, U). If the last letter also has to be a vowel, then it can be any one of the remaining 3 vowels. After we have selected the two vowels for the first and last positions, we have 6 letters left, of which 2 repeat. Thus, the number of ways to arrange these in-between letters is 6!/2! = 720/2 = 360. Finally, the number ways the words can be formed is 4 x 360 x 3 = 4,320.

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Re: In how many ways can you arrange the letters "APTITUDE" so as to form  [#permalink]

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11 Aug 2019, 14:43
The way I did was a bit longer than some answers, but helped me understand better the problem.

First I thought there are 8 letters in total and marked on the paper 8 spaces for the counting. The first letters will be able to use 4 potential letters, and the last one, 3 remaining ones, since there are 4 vowels.

Then, for the second space, I got 8 -2 (total - 2 vowels used) = 6, for the 3rd space I got 5 and so on. After multiplying it all, I got (120 x 72), but, since T is counted twice, I divided by 2 to remove double counting, which resulted in 60 x 72 = 4,320
Re: In how many ways can you arrange the letters "APTITUDE" so as to form   [#permalink] 11 Aug 2019, 14:43