Dave, Maria, Kate, Eddy, Donna and Josh go to a movie and

Question

Dave, Maria, Kate, Eddy, Donna and Josh go to a movie and sit next to each other in 6 adjacent seats in the front row of the theatre. If Josh and Donna will not sit next to each other, in how many different arrangements can the six people sit?

Will post OA later.

omomo · Answer

OA: 480 
Dave, Maria, Kate, Eddy, Donna and Josh go to a movie and sit next to each other in 6 adjacent seats in the front row of the theatre. If Josh and Donna will not sit next to each other, in how many different arrangements can the six people sit? 

Here's the explanation (But I think there's a quicker way to solve this problem...anyone knows?):

Step 1: Draw a diagram to show the row of 6 chairs. Begin with the people involved in the constraint.
Let's begin with Josh. If Josh is the first to sit, he can sit in any of the 6 seats.

Step 2: Donna has 3 or 4 possbible choices. 
If Josh has chosen to sit in one of the two ends (1 or 6), only one seat is off-limits to Donna-the seat immediately adjacent to Josh. This leaves 4 remaining seat options for Donna.
If Josh has chosen to sit in one of the four middle seats (2,3,4, or 5), exactly two seats are off-limits to Donna-the two seats on either side of Josh. This leaves 3 remaining seats options for Donna. 

Thus, in 1/3 of the cases, Donna will have 4 seats options and, in 2/3 of the cases, Donna will have only 3 seats options.

After seating Josh and Donna, seat the non-constrained people.

Dave has 4 possible choices.
Maria has 3 possible choices.
Kate has 2 possible choices.
Eddy has 1 possible choice.

To compute the total number of permutations, find the product of the number of choices for each of the 6 people:
6 * (1/3*4 + 2/3*3) * 4 * 3 * 2 * 1 = 480

There are 480 different ways in which Josh and Donna will NOT sit next to each other.

qpoo · Answer

total 6P6 ways to sit 6 people, take out the possible ways Josh and Donna
sitting together 2x(5P5)

6P6 - 2x(5P5) = 480

sudhagar · Answer

qpoo: Could you pls explain how you got 5P5?

qpoo · Answer

since Josh and Donna are sitting together, we can view them as 'one entity',
so we have 5 such entities to arrange. For each such arrangement, we can have (Josh, Donna) and (Donna, Josh), so 5P5 x 2

bsmith75 · Answer

I got 480 by drawing the chairs and created the equation:

2 * (4 * 4!) + 4*(4 * 3 * 3!)
2(96) + 4(72) = 480

Using the same logic as omomo w/ sitting on the ends there are 4 possible people that can sit there and 4! for the remaining seats. 

On the interior seats there are 4 possibilities on one side and 3 on the other leaving 3! to fill the remaining. You multiply this times four since 4 seats fit this description.

I like qpoos methodology though, very slick.

B.

sudhagar · Answer

Thanks qpoos. I missed the point that when J & D sit together there are 4! ways that other people could be seated :-(

ywilfred · Answer

The total number of ways to seat 6 people = 6!/0! = 720 ways (permutation as order matters here)

The only was Josh and Donna can seat next to each other is if they take seat (1,2), (2,3), (3,4), (4,5) or (5,6). For each seating combination (1,2), there are 2 ways to seat Josh and Donna. For this case, Josh in seat 1 and Donna in seat 2 or Josh in seat 2 and Donna in seat 1.

Also for each case, there are 4! ways to seat the rest of the people.

So total number of permutations Josh and Donna will seat next to each other = 4! * 5 *2 = 240 ways

So number of ways they will not seat next to each other = 720-240 = 480 ways

arichards728 · Answer

Total ways to seat 6 people = 6!

Total way to sit Josh and Donna next to each other - compare them as one person.

JD 5 * 4 * 3 * 2 * 1 But we must also realize that the order for D and J can be switched to DJ (*2!)

6! - 5!2! = 480

Dave, Maria, Kate, Eddy, Donna and Josh go to a movie and

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