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# What percent of the different arrangements of the letters of

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Senior Manager
Joined: 21 Oct 2013
Posts: 419
What percent of the different arrangements of the letters of  [#permalink]

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30 Jul 2014, 02:13
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Difficulty:

55% (hard)

Question Stats:

60% (01:24) correct 40% (01:42) wrong based on 219 sessions

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What percent of the different arrangements of the letters of the word ABACUS are those in which the vowels appear together?

A. 10%
B. 20%
C. 40%
D. 50%
E. 60%
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Re: What percent of the different arrangements of the letters of  [#permalink]

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30 Jul 2014, 02:22
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The total number of different arrangements are 6!/2=360. We divide here by 2, since we have two same letters A.

Letters A, A, and U are vowels. Put them together and consider as one object. So, we have 4 different objects (AAU) B C S. The number of different arrangements of them are 4!=24. And we have 3 possibilities to arrange vowels: AAU, AUA, UAA. Therefore, we have 24*3=72 different arrangements.

Hence, 72/360*100%=20%.

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Re: What percent of the different arrangements of the letters of  [#permalink]

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30 Jul 2014, 02:27
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goodyear2013 wrote:
What percent of the different arrangements of the letters of the word ABACUS are those in which the vowels appear together?

A. 10%
B. 20%
C. 40%
D. 50%
E. 60%

The number arrangements of ABACUS is 6!/2! = 360 (there are 6 letters in this word out of which there are two A's).

Consider vowels A, A, and U as one unit: {AAU}. So, we'll have total of 4 units {AAU}{B}{C}{S}, which can be arranged in 4! ways. A, A, and U within their unit can be arranged in 3!/2! ways (because of duplicate A's again). Hence the number of arrangements when the vowels are together is 4!*3!/2! = 72.

Percentage = 72/360*100 = 0.2.

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Re: What percent of the different arrangements of the letters of  [#permalink]

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01 Aug 2014, 00:06
2
goodyear2013 wrote:
What percent of the different arrangements of the letters of the word ABACUS are those in which the vowels appear together?

A. 10%
B. 20%
C. 40%
D. 50%
E. 60%

Number of Possible Arrangements of Abacus:$$6!$$
Number of Arrangements of the vowels being together: $$3!$$
Number of Arrangements of the vowels together with the 3 other consonants: $$4!$$

$$\frac{4!*3!}{6!}=1/5=20%$$

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Re: What percent of the different arrangements of the letters of  [#permalink]

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04 Aug 2015, 11:49
1
goodyear2013 wrote:
What percent of the different arrangements of the letters of the word ABACUS are those in which the vowels appear together?

A. 10%
B. 20%
C. 40%
D. 50%
E. 60%

First, let's determine the number of total possibilities in arranging the letters. There are six spaces, so the total number of arrangements is 6!, or 360.
Next, we need to figure out how to determine the number of ways that we can arrange the 3 vowels together - simply place them together (as in AAU) and call that a single place.
Next, we must determine the number of ways to arrange the now 4 units (i.e., AAU, B, C, S). Like above, there are 4 units and 4 places so the number of arrangements is 4!, or 24.
Finally, we need to account for the number of ways we can arrange AAU. We can either write out each unique iteration (AAU, AUA and UAA) or calculate as 3!/2! and get 3.

Putting this all together, we get the number of ways to arrange the letters so that the vowels are together is 4! x 3 ==> 72
the number of total arrangements of all the letters is 6! ==> 360

72/360 = 1/5, or 20% Correct answer is B
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Re: What percent of the different arrangements of the letters of  [#permalink]

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03 Nov 2017, 21:07
goodyear2013 wrote:
What percent of the different arrangements of the letters of the word ABACUS are those in which the vowels appear together?

A. 10%
B. 20%
C. 40%
D. 50%
E. 60%

The leftmost arrangement satisfying the constraints is AAU---

The number of permutations can be found using NC*C*(R+1)
where NC is the number of permutations of non constraint elements and c is the number of permutations of constraint elements and R is the number of elements after the rightmost constraint element= 3!*(3!/2!)*4=72
Total number of combinations = 6!/2!=360

Percentage = (72/360 )*100=20%. Hence B

Note: There are nuances in using the above formula which I will explain in the appropriate questions.
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Re: What percent of the different arrangements of the letters of  [#permalink]

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06 Nov 2018, 08:19
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