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I have one question that bothers me. The total number of words that can be made by writing the letters of the word PARAMETER so that no vowel is between two consonants. The answer is 1800. I couldn't solve it. Please help.

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closed as off-topic by user99914, dustin, Eric Stucky, Venus, colormegone Jan 20 '15 at 7:28

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  • $\begingroup$ (Sorry, my vote was incorrect, this is not a particularly good example of an abstract duplicate of the MISSISSIPPI question.) $\endgroup$ – Eric Stucky Jan 20 '15 at 7:18
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The consonants are P,R, M, T, R. The vowels are A, A, E, E. Group all the consonants as one symbol. Then the number of distinct words you can make out with this symbol and the $4$ vowels are $$\frac{5!}{2! 2!}=30$$ But within this symbol you can create new words by rearranging the consonants. This can be done in $$\frac{5!}{2!}=60$$

Thus the total number of words formed is $1800$.

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OK. The word must consist of the consonants being in a single block and the vowels flowing around them. So the answer will be $$\text{consonant arrangements}\times\text{vowel arrangements}\times\text{spots the consonants can start}$$ Which will be $$(\frac{5!}{2!})(\frac{4!}{2!2!})(5)=1800$$

5 spots that the consonants could start are $$CCCCCVVVV\\VCCCCCVVV\\VVCCCCCVV\\VVVCCCCCV\\VVVVCCCCC$$

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