benjiboo wrote:

swatirpr wrote:

benjiboo wrote:

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction? Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them.

How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person? The math/strategy behind the second part would really be appreciated.

Second Part:

Seat

1 ---- 2 ---- 3 ---- 4 ---- 5 ---- 6

R ---- ---- T ---- ---- ---- ---- ---- = 2*4*3*2*1 = 48

---- --R ---- ---- --T ---- ---- ---- --= 2*4*3*2*1 = 48

---- ---- --- R---- ---- --T---- ---- -= 2*4*3*2*1 = 48

---- ---- ---- ---- -R---- ---- --T----= 2*4*3*2*1 = 48

So Total ways 48+48+48+48 = 192

Explain please. Thanks!

Second Part:

condition - Ron and Todd separated by exactly 1 person

So If R @ 1 then T @ 3 - 1st way

If T @ 1 then R @ 3 - 2nd way

So R n T can sit 2 ways

For remaining Seats 2, 4, 5, 6, 4 people can seat 4*3*2*1 ways

Total for this seating arrangement 2*4*3*2*1 =48

OR

If R @ 2 then T @ 4 - 1st way

If T @ 2 then R @ 4 - 2nd way

So R n T can sit 2 ways

For remaining Seats 1, 3, 5, 6, 4 people can seat 4*3*2*1 ways

Total for this seating arrangement 2*4*3*2*1 =48

OR

If R @ 3 then T @ 5 - 1st way

If T @ 3 then R @ 5 - 2nd way

So R n T can sit 2 ways

For remaining Seats 1, 2, 4, 6, 4 people can seat 4*3*2*1 ways

Total for this seating arrangement 2*4*3*2*1 =48

Sitting arrangement

1 ---- 2 ---- 3 ---- 4 ---- 5 ---- 6

R ---- ---- T ---- ---- ---- ---- ---- = 2*4*3*2*1 = 48

---- --R ---- ---- --T ---- ---- ---- --= 2*4*3*2*1 = 48

---- ---- --- R---- ---- --T---- ---- -= 2*4*3*2*1 = 48

---- ---- ---- ---- -R---- ---- --T----= 2*4*3*2*1 = 48

So Total ways 48+48+48+48 = 192

Please let me know if I am doing this wrong.