How many 5-letter “words” can be formed from the letters of the word FORMULATED, if each must contain 2 vowels and 3 consonants?

Question

How many of these will have the vowels at the two ends on the “word”?

Answer 1

First, observe that FORMULATED has 10 letters; 4 vowels and 6 consonants [note: no duplicates].

The question, “’How many of these have the vowels at the two ends on the ‘word’?” is an excellent starting point. It means that the “word” looks like: v c c c v. These vowel positions are fixed and characters do not repeat, so we have 4*3 (that is, 4C2) possibilities for the vowels, and 6*5*4 (that is, 6C3) possibilities for the consonants. That’s a total of (4*3)*(6*5*4) = 1440 “words” that have a vowel on each end.

The reason that this last question is such a valuable clue is that it is really asking, “How many ‘words’ have two vowels in two specific positions?” Now, if we know how many possible two-positions there are in a 5-letter word, we simply multiply (because there will be 1440 for each of them). The possible positions are:

c c c v v
c c v c v
c c v v c
c v c c v
c v c v c
c v v c c
v c c c v
v c c v c
v c v c c
v v c c c

Wow! That’s merely 5C2 (that is, choose two of the five positions to be a vowel) = 10.

There are 1440*10 = 14400 possible 5-letter “words” with two vowels and three consonants.

Answer 2

An interesting problem.

First, let's choose two vowels, which can be done in 4C2 = 6 ways.

Next, let's choose three consonants, which can be done in 6C3 = 20 ways.

 

To reflect all possible combinations of two vowels and three consonants, we shall multiply: 120 such choices.

But these don't end the problem. Now that we have any particular combination of the vowels & consonants, these five things can be arranged in 5! orders.

 

My final answer is therefore 120(5!).  Finish the job, if you agree with the reasoning.