# How many arrangements of the letters in INSTRUCTOR have all of the following properties simultaneously?

How many arrangements of the letters in INSTRUCTOR have all of the following properties simultaneously?

(a) The vowels appearing in alphabetical order

(b) At least 2 consonants between each vowel

(c) Begin or end with the 2 Ts (the Ts are consecutive)

I tried:

We first place the vowels, IOU and there's only 1 way to do that. Then, we place TT and there's 2 ways to do that:

IOUTT or TTIOU

Now we're left with 5 consonants RRNSC. This is where I'm really having trouble with the question.

These are the possible places they can go:

_ I _ _ _ O _ _ _ U _ TT _

There must be 2 consonants between I and O and O and U. But there can't be more than 3 consonants between either I and O or O and U.

I don't know what to do from here. Do I split it into cases?

• Splitting carefully into cases is certainly possible, but there could be a better way... – true blue anil Jan 25 '16 at 17:28

There are rules for the TTIOU and then there are "five blank slots" for the remaining five letters. Where can we put the five blank slots and where can we put the TTIOU

There are 10 places total. The first 2 or the last 2 are occupied by the TT. (So that's 2) That leaves 8 places for the IOU. The I _ _ O _ _ U span seven of eight slots. We need to add an extra slot either before the I, between the I and the O, between the O and the U, or after the U. (That's 4)

So there are 2 * 4 = 8 ways to place the five blank slots.[*]

The ways to arrange the remaining 5 letters is 5!/2 =60 into the blank slots. (5 choices for the N, four choices for S once we fixed where we put the N, ..etc. = 5! ways to place 5 distinct letters, but the R's are interchangable o 5!/2)

So total number of ways is 8*60 = 480.

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[*]If that wasn't clear they are:

TTI_ _ O _ _ U _ I_ _ O _ _ U _ TT

TT _ I _ _ _ O _ _ U _ _ I _ _ _ O _ _ U _ TT

TT_I _ _ O _ _ _ U _ _I _ _ O _ _ _ U _TT

TT_I _ _ O _ _ U _ _ _I _ _ O _ _ U _ _TT

$1.$ There are $2$ ways of placing $TT,$ and we shall simultaneously need to divide the permutations obtained by $2$ due to two $R's,$ which cancel out, so we can ignore $TT$ and assume $5$ distinct consonants, and permute them in $5!$ ways.

$2.$ Two consonants between vowels is mandatory,
the extra $5th$ one has just $4$ locations to go to: $\;\;\uparrow I\uparrow O\uparrow U\uparrow$

$3.$ Thus ans $= 4*5!$