How many arrangements of the word "instructor" have three consecutive vowels?

Answer 1

- IOU permutations = 3! = 6 possibilities
- Position of first vowel = 8 possibilities
- Pick two spots out of remaining 7 for the two R's = 7C2 = 21 possibilities
- Pick two spots out of remaining 5 for the two T's = 5C2 = 10 possibilities
- Place the N, S and C in the remaining 3 spots = 3! = 6 possibilities.

Total = 6 * 8 * 21 * 10 * 6 = 60,480.

Maths Rocks: You just need to divide your answer by 4, to account for the two T's and two R's.

Answer 2

I don't think anyone quite has it yet.

There are 3 vowels, which can be placed in 3! orders.
There are then 7 consonants. Rather than use 7!, consider the block of 3 vowels to fill a slot too. Then there are 8! ways of arranging 7 consonants and one block of 3 vowels in order.
Finally, there are 2 repeated letters, so you need to divide by 2!*2!

3!*8!/(2!*2!) = 60480

Answer 3

instructor = 10 letters with t for 2 times
3 vowels can be arranged in factorial(3) = 6 ways
7 consonants can be arranged in factorial(7) / factorial (2) ways
And these 2 groups can be arranged in factorial (2) ways

Finally
6 x factorial (7) / factorial (2) x factorial (2)
= 6 x factorial (7) ways

Answer 4

iou together=3!
total arrangements=3!x8!
thnx 4 pointing out the mistake

Answer 5

You can have the I, O and U in six different combinations... IOU, IUO, OIU, OUI, UIO, UOI.

The first can start the word, be in second position, etc... there are 8 possibilities for this.

8*6 = 48.