Apples: In how many ways can the apples be placed in a row?

In how many ways can 3 Granny Smith, 4 red Delicious and 5 Macintosh apples be put on a shelf if 2 particular ones must be in the middle, if two particular ones must be sepreated and if each type is placed together.

I don't think I am on the right track, but I gave it a shot.

my solution:

treat 2 apples as one, (12-1)! = 11! = 39916800

total apples - 2 * (answer from a)
12! - 2(39916800) = 39916800

3! * 4! * 5! = 17280

This problem is very much ill-defined!
Which two? Do we take cases? “Two Granny’s” then “two Delicious” then “two Macintosh”?
Are you saying that the apples are all different?
Is there more to the actual problem that you posted?

No there is no more to the actual problem. It doesn't specify which two, so I would assume you can use any two.

Re: Apples: In how many ways can the apples be placed in a r

Clifford said:

In how many ways can 3 Granny Smith, 4 red Delicious and 5 Macintosh apples be put on a shelf if 2 particular ones must be in the middle, if two particular ones must be sepreated and if each type is placed together.

Click to expand...

Please answer this question: Why are you assuming that all the apples are different?
Why would the question bother to give a breakdown of types if that were the case.
Rather, I infer from the wording that the Granny Smith’s are identical and the same of the other two types.
In effect we have GGGDDDDMMMMM to arrange.

I have no idea why I am assuming they are different, since they shouldn't be.

would this be a better approach then:
treat 2 granny smiths as one, giving us 2 granny smiths, 4 declious and 5 macs.
11! / 2! 4! 5!

part 2:
12! / 3! 4! 5! - 2 (answer from a)

part 3:

3! * 4! * 5!

If I were you, I would give this answer the question this way:
“This question is so very ill-defined that no answer is possible.
There is no way that any competent mathematician would written this problem."
If you do that, I will back you up.