In how many ways can the word Josephite be arranged if the first letter is a vowel?

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tawhid

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Hey,I'm pretty new to combinatronics, forgive me for my naive question.

In how many ways can the word Josephite be arranged if the first letter is a vowel?

I'm kinda confused, if there repetitions aren't counted then,there are 3 unique vowels and 8 remaining letters so arrangements shall be 3*8! Right?
 
tawhid said:
Hey,I'm pretty new to combinatorics, forgive me for my naive question. I'm kinda confused, if there repetitions aren't counted then,there are 3 unique vowels and 8 remaining letters so arrangements shall be 3*8! Right?
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Is that your final answer, or just a first step?

It seems to me that there would be two cases.
  • How many ways are there to arrange the word if the first letter is O or I?
  • How many ways are there to arrange the word if the first letter is E?
By the way, there are not three unique vowels; there are three distinct vowels. One of them is not unique! At least, that's how I use the words, which are easily confused.
 
Thanks for the clarification haha. Distinct seems like the best in this case. Yes it is my final answer. As it seems like it satisfies both cases.
 
tawhid said:
Thanks for the clarification haha. Distinct seems like the best in this case. Yes it is my final answer. As it seems like it satisfies both cases.
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I thought I strongly hinted that you should think again. Did you?

When I'm not sure of my answer to a problem like this, I sometimes try a smaller version of the problem that I can check by actually counting. I suggest you try this with the word OGEE. How many arrangements do you calculate that start with a vowel? After you do that, then try actually listing them (it's a pretty short list). Compare, and see if your thinking worked. Then think again.

And then show us your example and your new work, because there will be more to discuss.

My general rule for most combinatorics problems (certainly for any interesting ones!) is: Think Again! That is, try solving two different ways, and if they disagree, try to fix them both.
 
tawhid said:
In how many ways can the word Josephite be arranged if the first letter is a vowel?
Click to expand...

The question should be "In how many ways can the letters of the word "Josephite" be arranged if the first letter
is a vowel?"

Also, try not to put the whole question in the headline. For instance, write "Number of ways to rearrange letters"
or some similar shortened phrase to let people know what the problem will be generally concerning. Then, write
out the full question at or near the beginning of the body of the post.
 
In how many ways can you arrange n distinct letters? Now try...
If the first letter is O, then In how many ways can the letters of the word Josephite be arranged?
If the first letter is i, then In how many ways can the letters of the word Josephite be arranged?
If the first letter is e, then In how many ways can the letters of the word Josephite be arranged?
 
Steven G said:
In how many ways can you arrange n distinct letters? Now try...
If the first letter is O, then In how many ways can the letters of the word Josephite be arranged?
If the first letter is i, then In how many ways can the letters of the word Josephite be arranged?
If the first letter is e, then In how many ways can the letters of the word Josephite be arranged?
Click to expand...
I would change this a little:

In how many ways can you arrange n letters, not all distinct? Now try...​
​
If the first letter is O, then In how many ways can the letters of the word Jsephite be arranged following that?​
​
If the first letter is i, then In how many ways can the letters of the word Josephte be arranged following that?​
​
If the first letter is e, then In how many ways can the letters of the word Josephit be arranged following that?​
 
If the first letter is O, then the remaining 8 letters of the word Jsephite can be arranged in 8! ways but there are 2 'E's, so it is [imath]\frac{8!}{2!}[/imath]
For I the ways are same [imath]\frac{8!}{2!}[/imath]
but since one E is at first, there are now 8! ways to arrange the remaining 8 letter.
 
tawhid said:
If the first letter is O, then the remaining 8 letters of the word Jsephite can be arranged in 8! ways but there are 2 'E's, so it is [imath]\frac{8!}{2!}[/imath]
For I the ways are same [imath]\frac{8!}{2!}[/imath]
but since one E is at first, there are now 8! ways to arrange the remaining 8 letter.
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So the final answer is?

For OGEE, the number by the same reasoning is [imath]\frac{3!}{2!}+3!=9[/imath], and the arrangements are
  • OGEE
  • OEGE
  • OEEG
  • EOGE
  • EOEG
  • EGOE
  • EGEO
  • EEGO
  • EEOG
 
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