Confusion about permutations of "HOME" with vowels together

Question

I have a confusion regarding a question: we are given the word "HOME" and we have to arrange its letters so that the letter "H" always comes first. My concept is to take four chairs on which these four persons are to be seated so that person "H" always sits on first place and if H sits only on first chair then we are left with three chairs for three persons and we can arrange them in 3! ways so the answer is in "6 ways" (I got right answer with this concept).

Now the second question is if I have a word "HOME" and we have to arrange it in some ways so that vowels always come together. now I apply the above chairs concept in this question I have four chairs and O, E (vowels) always sit together. So they can sit in 2! ways on two and we are left with 2 chairs so we can arrange the remaining two persons in two ways so according to this concept my final answer is 2!x2!=4 ways but the correct answer is "12 ways"

Please clear why I can't apply my chairs concept in the second question or please tell me if I am wrong. Thanks

Answer 1

You missed selecting the pair of chairs that OE sit on. The leftmost vowel can be on any one of three chairs, which is the factor three you are missing.

Answer 2

i have a confusion regarding a question we are given a word "HOME" and we have to arrange it in some ways so that the letter "H" always come on the first place. my concept is i take four chairs on which these four persons are to be seated so that person "H" always sit on first place and if H sit only on first chair then we are left with three chairs for three persons and we can arrange them in 3! ways so the answer is in "6 ways".( i got right answer with this concept)

Yes.   You are arranging four symbols so that one specific symbol is in a fixed position.   Just count the ways to put $H$ in first place, then to arrange the remaining three symbols in the last three places.   Thus $1!\times 3!$ ways ($6$).

$$\rm H\,OME\,, H\,OEM\,, H\,EOM\,, H\,EMO\,, H\,MEO\,, H\,MOE$$

Now the second question is if i have a word "HOME" and we have to arrange it in some ways so that vowels always come together now i apply the above chairs concept in this question i have four chairs and O,E (vowels) always sit together. so they can sit in 2! ways on two and we are left with 2 chairs so we can arrange the remaining two persons in two ways so according to this concept my final answer is 2!x2!=4 ways but the correct answer is "12 ways"

Now you are arranging four symbols so that two specific symbols are always adjacent.   Here you must count the ways to arrange the two symbols to stick together into one composite symbol, then to arrange that with the two remaining (so three symbols: "$H$", "$M$", "$O\!\!\!E$").   There are $2!$ ways to do the first task, and $3!$ ways to do the second.   So $12!$ ways.

$$\rm H\,M\,O\!\!E\,, H\,O\!\!E\,M\,, O\!\!E\,H\,M\,, M\,H\,O\!\!E\,, M\,O\!\!E\,H\,, O\!\!E\,M\,H \\ H\,M\,E\!\!O\,, H\,E\!\!O\,M\,, E\!\!O\,H\,M\,, M\,H\,E\!\!O\,, M\,E\!\!O\,H\,, E\!\!O\,M\,H$$