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25 Apr 2006, 18:45
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If A,b,c,d,e sit randomly in a row, what is the probablity that a and b are not next to each other?
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25 Apr 2006, 18:50
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Number of ways to arrange the people = 5! = 120 (or 5P5)
The different ways a and b can be seated next to each other is:
a,b,_,_,_ ----> 3! ways
_,a,b,_,_ ----> 3! ways
_,_,a,b,_ ----> 3! ways
_,_,_,a,b ----> 3! ways
Total = 4*3! = 24 ways.
But this only considers a,b in this order. We need to also consider b,a as well. So this is another 24 ways. The number of ways a and b are seated next to each other is therefore 48.
P(a and b are not seated next to each other) = 1 - 48/120 = 3/5
The different ways a and b can be seated next to each other is:
a,b,_,_,_ ----> 3! ways
_,a,b,_,_ ----> 3! ways
_,_,a,b,_ ----> 3! ways
_,_,_,a,b ----> 3! ways
Total = 4*3! = 24 ways.
But this only considers a,b in this order. We need to also consider b,a as well. So this is another 24 ways. The number of ways a and b are seated next to each other is therefore 48.
P(a and b are not seated next to each other) = 1 - 48/120 = 3/5
25 Apr 2006, 19:02
Kudos
Bookmarks
I just wanted to add one more way of solving this.
The number of ways to arrange 5 poeple = 5! = 120 (Or 5P5) (This will always be the same, no matter what method you use)
Then let's consider a and b as 1 entity, so now we need to arrage (ab),c,d,e and the number of ways to do this is 4! = 24. Again, we need to consider reverse order, ba. So total = 48 ways.
Probability = 1-48/120 = 3/5
The difference in this method is that instead of writing down possible seating arrangements, we can just lump a and b together to be considered as 1 entity. It's faster and less error-prone.
The number of ways to arrange 5 poeple = 5! = 120 (Or 5P5) (This will always be the same, no matter what method you use)
Then let's consider a and b as 1 entity, so now we need to arrage (ab),c,d,e and the number of ways to do this is 4! = 24. Again, we need to consider reverse order, ba. So total = 48 ways.
Probability = 1-48/120 = 3/5
The difference in this method is that instead of writing down possible seating arrangements, we can just lump a and b together to be considered as 1 entity. It's faster and less error-prone.
