Number of permutations of a row of students.

How many ways can a photographer arrange six students in a row if:
a.) Two of the students (Ben and Greg) cannot be next to each other.
b.) Ben is somewhere to the left of Greg.

Gladius said:

How many ways can a photographer arrange six students in a row if:
a.) Two of the students (Ben and Greg) cannot be next to each other.
b.) Ben is somewhere to the left of Greg.

Click to expand...

Consider cases.

If Ben is on the far left, then Greg must be to his right. In how many ways can the students be seated, given that Greg cannot be in the second seat (counting from the left)?

If Ben is on the far right (that is, in the sixth seat), Greg cannot be in the fifth seat. This case will have as many options as the previous one.

If Ben is in the second seat, then Greg cannot be in the first or third. In how many ways can the students be seated in this case?

What about when Ben is in the third seat? :wink:

Gladius said:

How many ways can a photographer arrange six students in a row if:
a.) Two of the students (Ben and Greg) cannot be next to each other.
b.) Ben is somewhere to the left of Greg.

Click to expand...

The other four students sever as separators for Ben from Greg.
Thus, for a), there are 5=4+1 places to put Ben and Greg: \(\displaystyle {5 \choose 2}(2)(4!)\) to separate the two. WHY?

For part b) the answer is \(\displaystyle {5 \choose 2}(4!)\). Why in the world is that?

Hello, Gladius!

How many ways can a photographer arrange six students in a row if:

(a) Two of the students (Ben and Greg) cannot be next to each other.

Click to expand...

With no restrictions, there are: \(\displaystyle 6! = 720\) possible arrangements.

Let us count the number of ways that Ben and Greg are together.

Duct-tape Ben and Greg together.
So we have 5 "people" to arrange . . . There are: \(\displaystyle 5! = 120\) ways.

But Ben and Greg can be taped in two ways: \(\displaystyle BG\) or \(\displaystyle GB\).

Therefore, there are: .\(\displaystyle 2 \times 120 \:=\:240\) ways.

(b) Ben is somewhere to the left of Greg.

Click to expand...

There is a back-door approach to this problem.

There are 720 possible seating arrangements.

In half of them Ben is to the right of Greg
. . and in half of them Ben is to the left of Greg.

\(\displaystyle \text{Therefore, Ben is to the left of Greg in: }\:\tfrac{1}{2}\times720 \:=\:360\text{ arrangements.}\)