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
Unable to visualize a specific permutation problem
This topic has expert replies
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!
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!