Ways to arrange the letters in "TRIANGLE" if vowels must appear in alphabetical order?

Question

Just the question above, I saw it in my textbook and went a little crazy trying to solve it. So I really have no clue. Thanks.

Answer 1

Hint: Set the problem up as follows. $$\quad\cdot\quad A\quad\cdot\quad E\quad\cdot\quad I\quad\cdot$$ Wherever there is a dot, you can put some letters (from the remaining selection, TRNGL). How many ways can this be done? For $T$, you have 4 possible choices. For $R$ you then have 5 possible choices (either side of $T$ gives the additional choice). Repeating this process, gives the answer.

Answer 2

Hint:

First pick which locations will be occupied by vowels. After having chosen the locations, the order in which the vowels occupy those locations can only occur in one way as per the condition that the vowels must be in alphabetical order.

Next, in the remaining unused locations, place the consonants.

Answer 3

There are two respected solutions at a introductory level for this style of problem.
Solution 1: Consider what order the vowels appear in the word. In this case, vowels can appear in the word in 3P3 = 6 distinct ways. Since it can be reasonably assumed that each permutation of the order of the vowels is equally likely to happen, (This assumption is widely accepted in many materials), and only one of them, AEI, is alphabetical, that means a 1/6 of the total permutations have an alphabetical vowel ordering.
Solution 2: Choose 3 possible spots for the vowels. thats 8C3 possible spots that vowels may be in. Because for every spot, there is only 1 permutation of vowels that is alphabetical, our solution is just 8C3 * 5P5 (we calculate the total permutations (of consonants) for each satisfactory order.) Note the first solution is quicker.