# How many arrangements of INSTITUTIONAL have all of the following properties simultaneously?

How many arrangements of INSTITUTIONAL have all of the following properties simultaneously: a. No consecutive T's b. The 2 N's are consecutive c. Vowels in alphabetical order

I'm studying for a test and am pretty sure this question is going to stump me. I know I need to handle the N's as a single letter (NN), but I'm not sure how to handle the alphabetical order of vowels.

Any help is much appreciated, thanks.

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Removed the [probability] tag since the question has nothing to do with probabilities.. – Mårten W Apr 12 '13 at 11:53
Thanks for the help – CSDUDE Apr 12 '13 at 12:50

First place the vowels in alphabetical order.

$$AIIIOU$$

We now have $3$ $T$'s, $1$ $S$, $1$ $L$ and the $2$ $N$'s (which we shall treat as a group as you suggest).

We will first place the group of $N$'s, the $S$ and the $L$ as they have no restrictions on them.

There are $7$ gaps we can put the $S$ in.

There are $8$ gaps where we can then put the $L$.

Then there are $9$ gaps where we can put the two $N$'s. (sorry for forgetting this last time perhaps confusing you).

Then $10 \cdot 9 \cdot 8$ ways to place the $T$'s. But we have to divide this number by $3!$ as the $T$'s are not distinguishable. So $5 \cdot 3 \cdot 8=\binom{10}{3}\binom{10}{7}$ is the actual number.

So our final result is $7 \cdot 8 \cdot 9 \cdot (5 \cdot 3 \cdot 8)=60480$.

I'm SO sorry for getting it wrong last time. I have a weakness for overlooking things in combinatorics :).

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You should edit your original answer rather than posting a new one. – Alex Becker Apr 30 '13 at 5:12