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

_________________

Dennis

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