How many permutations of the word "TEACHER" are there if the vowels must appear in alphabetical order?

The word "BANANA" has 6 letters:
B: 1 time
A: 3 times
N: 2 times

The formula for permutations with repeated letters is $\frac{n!}{p_1! p_2! ... p_k!}$.
Here, $n=6$, A is repeated 3 times ($p_1=3$), and N is repeated 2 times ($p_2=2$).

Treat "NN" as a single block.
The items to arrange are now: B, A, A, A, (NN).
The total number of items to arrange is 5.
The letter A is repeated 3 times.

The number of permutations of the word "BANANA" where the two N's cannot be adjacent is $\boxed{\text{40}}$.

The word "TEACHER" has 7 letters.
Vowels: A, E, E (3 vowels)
Consonants: T, C, H, R (4 consonants)

The vowels A, E, E must appear in alphabetical order. This means their sequence must be A, E, E.

There are 7 total positions for the letters. We need to choose 3 of these positions for the vowels.
The number of ways to choose 3 positions out of 7 is given by the combination formula $\binom{n}{k}$:

Once the 3 positions are chosen, the vowels A, E, E must be placed in them in the specific alphabetical order (A, E, E). Since the two E's are identical, there is only 1 unique way to place them in the chosen slots (e.g., the first chosen slot gets A, the second gets E, the third gets E).

There are 4 remaining positions and 4 consonants (T, C, H, R). All these consonants are distinct.
The number of ways to arrange 4 distinct consonants in 4 distinct positions is $4!$.

Multiply the number of ways to choose positions for vowels, the number of ways to place vowels, and the number of ways to arrange consonants.

The number of permutations of the word "TEACHER" where the vowels must appear in alphabetical order is $\boxed{\text{840}}$.