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.
Best Answer
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 :).