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

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kirankp
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In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  02 Dec 2009, 01:24
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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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Bunuel
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  02 Dec 2009, 02:33
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prome2 wrote:
APTITUDE has 8 letters, so given that the letters start and end with a vowel there are 2 conditions.

1) There are 4 choose 2 possibilities for a vowel start/end combination. [Any 2 out of A,I,U,E]

2) The remaining 6 letters can be arranged in any order and still form a word. Thus permutate 6 objects.

Answer = (4!/2!2!)*6!= 4320

Edit : Given T is repeated and non vowel. I think the final answer should be divided by 2! to give 2160 (which is not an option)

Would love to get some advise!


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

Answer: C.

So I think the answer is right.You just didn't consider that order matters in the first case and repeated T in the second case, but these two mistakes compensated each other and you got the correct answer.

Hope it's clear.
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  02 Dec 2009, 01:52
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APTITUDE has 8 letters, so given that the letters start and end with a vowel there are 2 conditions.

1) There are 4 choose 2 possibilities for a vowel start/end combination. [Any 2 out of A,I,U,E]

2) The remaining 6 letters can be arranged in any order and still form a word. Thus permutate 6 objects.

Answer = (4!/2!2!)*6!= 4320

Edit : Given T is repeated and non vowel. I think the final answer should be divided by 2! to give 2160 (which is not an option)

Would love to get some advise!
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prome2
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  Updated on: 02 Dec 2009, 21:09
Crystal! Thanks Bunuel! +1

Originally posted by prome2 on 02 Dec 2009, 05:31.
Last edited by prome2 on 02 Dec 2009, 21:09, edited 1 time in total.
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chetan2u
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  17 Sep 2017, 04:25
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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] New post  18 Sep 2017, 04:08
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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

Answer C
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  22 Sep 2017, 07:28
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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.

Answer: C
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  11 Aug 2019, 15:43
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
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  29 Jan 2021, 15:08
Permutation with restrictions.

V _ _ _ _ _ _ V

4 <--- # of ways a vowel can be chosen for the first letter
3 <--- # of ways... last letter

4 x 3 = 12 possibilities

6 letters left...

6! / 2! = 360 <--- # of ways the remaining letters (including two vowels can be arranged)

360 x 12 = 4320 <--- # of possible words with vowels in first/last position

Answer is C.
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  13 Aug 2021, 07:18
Question -- why doesnt this method work:

8!/2! total ways to arrange the letters
A. ½ of the total will have a VOWEL as the first letter (since ½ of the total letters are vowels).
B. Similarly, ½ of the total will have a VOWEL as the last letter (since ½ of the total letters are vowels).

therefore we can take the total number and divide by 4-> ¼ of all will have a vowel at the beginning and end:

(8!)/(2!4)

Why is this flawed?
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Re: In how many ways can you arrange the letters "APTITUDE" so as to form [#permalink] New post  13 Aug 2021, 13:06
This question is tricky because, for example ATIUDETP is not a valid word and nor are many of the 4320 possible arrangements
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