probability

There are 6 students(A-F)standing in a row, find the probability thatany two of A,B and C do not stand next to each other.

Solution :
If the 6 students stand in a row without restriction :
Number of ways = 6P6

If any two of A, B and C do not stand next to each other :
Firstly, arrange D, E and F (denoted as X) in a row (3P3­): _X_X_X_
Then, Put A, B and C in any three spaces "_" (4P3).
Number of ways = 3P3 x 4P3

The required probability
= 3P3 x 4P3 / 6P6
= 3! x (4!/1!) / 6!
= 1/5


4 girls and 8 boys are arranged in a row at random.Find the probability that any 2 girls do not stand next to each other.

Solution :
If the 12 persons (4 girls and 8 boys) are arranged in a row withoutrestriction :
Number of ways = 12P12

If any 2 girls do not stand next to each other :
Firstly, arranged the 8 boys (B) in a row (8P8) :_B_B_B_B_B_B_B_B_
Then, Put the 4 girls in any 4 spaces "_" (9P4).
Number of ways = 8P8 x 9P4

The required probability
= 8P8 x 9P4 / 12P12
= 8! x (9!/5!) / 12!
= 14/55