Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is :

The value of (2.¹P₀ – 3.²P₁ + 4.³P₂ .... up to 51^th term)
+ (1! – 2! + 3! – ..... up to 51^th term) is equal to :

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :