SOLUTION: How many ways can 6 people be lined up to get on the bus. If certain 2 people refuse to follow each other how many ways are possible?

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Question 1003467: How many ways can 6 people be lined up to get on the bus.
If certain 2 people refuse to follow each other how many ways are possible?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There are 6%21=1%2A2%2A3%2A4%2A5%2A6=720 ways that 6 people can be lined up to get on the bus.
(That is because there are 6 ways to choose who is first,
for each of those choices, there are 5 ways to choose who is second,
and so on).

There are 4%21%2A2%2A5=%281%2A2%2A3%2A4%29%2A5=5%21=120 ways to line up 6 people placing together the 2 people who refuse to be next to each other.
{That is because there are 4%21 ways to arrange the other 4 people,
2 ways to arrange the 2 people who refuse to be next to each other,
and 5 places to insert the problem pair in the line formed by tho other 4).

Since 120 of the 720 ways that 6 people can be lined up to get on the bus place together the 2 people refuse to be next to each other,
there are 720-120=600 ways to line up the 6 people keeping apart the 2 people who refuse to be next to each other.