Problem Solving

Hi - I'm having difficulty in visualizing problems of the types where 3 or more elements do NOT have to be together in permutation problems:

Problem 1: How many ways can the letters of RAINBOW be arranged so that the vowels are never together?
or
Problems 2: How many ways can 6 students and 4 teachers be seated so that no 2 teachers are together?

Now, the answer explanation says that:

Problem 1 - 4 consonant can be arranged in 4! ways which leaves us with 5 places to place the 3 vowels - HOW 5 PLACES?? IT SHOULD BE 3 PLACES, RIGHT?

Problem 2- 6 students can sit in 6! ways, which leaves 7 places to seat the 4 teachers - AGAIN, HOW 7 PLACES??

Please help.
Tanvi

XCXCXCXCX

count number of X's in the above = 5


Same with the second problem.

Problem 2:

Let's say S = students and _ = empty place where we can put a teacher.

First, line up all the students so there's exactly one gap between each one of them for the teacher.

There are 6 students so:

S_S_S_S_S_S

This gives us 5 spots to put the teachers, right?

But think about the edges! There's 2 more spaces on the edges. If we place a teacher there he'll have an empty space on one side and a student on another which satisfies the condition.

_S_S_S_S_S_S_

So in total, we have 7 spots to put the teachers. Hope this helps!