What word has two vowels two consonants and two vowels all in a row

No,a vowel in the English language is a letter. There are five, a,e,i,o, and u. These letters are connectors for the consonants. Together, consonants and vowels, make up words. The word square has two vowels, a and e.

Consider a seven-letter word formed by mixing up the letters in the word COMBINE. How many ways can you do this if no vowel is isolated between two consonants?
(eg. EBMCION and MOIENCB are acceptable, but BEMCNIO is not)

The final answer is given to be $1872$.

We are being asked to find the numeric value of (Total number of seven-letter words) - (Number of words such that a vowel is isolated between two consonants). The word "COMBINE" has $4$ consonants and $3$ vowels. To compute the number of constructed words with an isolated vowel between two consonants, I tried the following:

1st attempt:

We may generalize the case, representing consonants as $1$s and vowels as $0$s. Now, we are finding the total number of seven-digit binary strings with four $1$s and three $0$s, containing the substring $101$. There are $\frac{4!}{2!2!} = 6$ strings that can be formed from the remaining two $0$s and $1$s. We may insert the substring $101$ into any position in these 6 strings. We have five available positions, because there are four digits. Therefore, there are $6 \times 5=30$ seven-digit binary strings with four $1$s and three $0$s, containing the substring $101$.

Since each vowel and consonant is unique in the word COMBINE, there are $30 \times 4!\times3!=4320$ seven-letter words formed from COMBINE that contain an isolated vowel between two consonants, if we account for permutations of the vowels and consonants. However, $7!-4320=720 \neq 1872$, meaning my answer is incorrect.

2nd attempt:

Since there are three vowels, for there to be an isolated vowel between two consonants, either all vowels are separated, or two vowels may be grouped up together while the remaining vowel is isolated (there is no other case).

Case when all vowels are isolated:
First, we arrange the consonants, Then, we choose three out of five possible places to place the vowels (.C.C.C.C., where C denotes a consonant and the periods represent possible places to put a vowel). We finally account for the arrangement of the vowels. $$ P(4,4) \times C(5,3) \times P(3,3) = 1440.$$

Case when one vowel is isolated, while the remaining two are a pair:
First, we arrange the consonants. We then choose a vowel. Then, we choose a position out of three possible places to put it (C.C.C.C). Then, we choose a position out of four possible places to place the pair of the remaining vowels (.CVC.C.C. just as a possible example). Finally, we account for the arrangement of the vowels in the pair. $$ P(4,4) \times C(3,1) \times C(3,1) \times C(4,1) \times P(2,2) = 1728.$$ $7!-(1440+1728) = 1872$, which confirms that this is the correct answer.

What is wrong with my first attempt? Where did I make my error?

If you picked the name of a U.S. state, at random, out of a hat, you would have slightly less than a one in four chance of choosing one that starts with a vowel. Twelve of the 50 United States have names that start with a vowel. But you’d only have a one in 50 chance of choosing one that starts with two consecutive vowels—because there’s only one. Any ideas which it is? Once you work out the answer, see if you can also figure out which letter never appears in any U.S. state name.

If you live in the midwest, this might have been a no-brainer—specifically, if you live in a certain state situated between the Missouri and Mississippi Rivers. That’s right, the answer is Iowa! Its name, of course, starts with an “I” followed by an “O”—no other state’s name begins with two vowels in a row. (Go ahead, test it out with every other state name that starts with a vowel!) Want to know how Iowa got that name? Here’s the story behind the origin of every U.S. state name.

And that’s not the only vowel-related bit of trivia that makes Iowa unique. It’s also the only state whose two-letter abbreviation consists of two vowels. That may not seem surprising—until you realize that its abbreviation is actually not the same two letters it starts with! The abbreviation for Iowa is “IA,” not “IO.” After all, Iowa does contain four letters, three of which are vowels. (The same is actually true for Ohio, but the distinction doesn’t carry over to its abbreviation, “OH.”)

What about on the other end of the words? Well, ending with two consecutive vowels is less of a distinction for U.S. state names—nine of them do it. In fact, it’s actually rarer for them to end with two consecutive consonants! (Seriously!) And if you think that’s interesting, feast your brain on the strangest fact about every U.S. state.

Originally Published: April 30, 2020

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