10 books are placed at random in a shelf. The probability that a pair of books will always be together is ?
Options:
(A) 1/8
(B) 1/4
(C) 1/5
(D) 1/9
(E) 1/1
Answer and stream of consciousness: (C)
Stream of consciousness: 10 books can be rearranged in 10! ways consider the two books taken as a pair then number of favourable ways of getting these two books together is 9! 2!
Required probability = 1/5

10 books are placed at random in a shelf. The probability that a pair of books will always be together is ?
Options:
(A) 1/8
(B) 1/2
(C) 1/5
(D) 1/5
(E) 1/3
Answer and stream of consciousness: (C)
Stream of consciousness: 10 books can be rearranged in 10! ways consider the two books taken as a pair then number of favourable ways of getting these two books together is 9! 2!
Required probability = 1/5

Pipe A can fill a tank in 5 hours, pipe B in 10 hours and pipe C in 30 hours. If all the pipes are open, in how many hours will the tank be filled?
Options:
(A) 6 hours
(B) 8 hours
(C) 3 hours
(D) 9 hours
(E) 1 hours
Answer and stream of consciousness: (C)
Stream of consciousness: Part filled by A + B + C in 1 hour = 1/5 + 1/10 + 1/30 = 1/3
All the three pipes together will fill the tank in 3 hours.

If a and b are unique prime integers, which of the following must be even?
Options:
(A) ab
(B) a2b
(C) 2a2b
(D) 2ab
(E) ab^2
Answer and stream of consciousness:
(C)
Stream of consciousness: Let's assume that a is an even prime integer and b is an odd prime integer, their subtraction results in an odd integer and an odd integer multiplied by an even integer yields an even integer.