From the word SHOWERDOWN, how many 4-letter words begin and end with a vowel

combinationscombinatoricspermutationssolution-verification

I had this question from my Probability worksheet:

From the word SHOWERDOWN, how many 4-letter words begin and end with a vowel (i.e. O, E, or O)?

I did:
Case 1: 2W's = 1
Case 2: All distinct letters = 6P2 = 30

I then multiplied 30+1 by $\frac{3P2}{2!}$, because I need to also arrange the vowels and divide out the repetition of 'O'.

However, I am unsure if my answer (i.e. 93) is correct, so any clarification would be much appreciated.

Best Answer

You are missing cases - for example, what about cases where all three vowels are present? Two of them are at two ends and one of them is one of the two middle letters.

Once you choose two vowels, you are left with $7$ distinct letters:
S H W R D N E or S H W R D N O

There are $ \displaystyle {7 \choose 2}$ ways to choose two letters and $2$ ways to arrange them. Also there are $3$ way to place two vowels:
O _ _ O, E _ _ O, O _ _ E

That leads to $~\displaystyle 3 \cdot 2 \cdot {7 \choose 2} = 126$ four letter words, not including words with two W's. There are $3$ four letter words with two W's.

So, the answer should be $129$.

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