Answer: B
Total no. of seats = 1 grandfather+ 5 sons and daughters + 8 grandchildren = 14.
The grandchildren can occupy the 4 seats on either side of the table in 4! = 24 ways.
The grandfather can occupy a seat in (5-1)= 4 ways (4 gaps between 5 sons and daughter).
And, the remaining seats can be occupied in 5!= 120 ways (5 seat for sons and daughter).
hence total number of required ways = 8! * 480.

Answer: C
Let total number of persons present in the party be x,
then x(x-1)/2 = 105
=> x=15.

Answer: D
Required number of ways = 3_C2 * 4! *4! = 1728

Answer: D
At least 1 question from each section is compulsory, so from the 1st section the candidate can attempt 1 or 2 or 3 or 4 questions. In each section each question can be dealt with in 2 ways, i.e either he attempts it or leaves it.
So far 4 question there are 2(2)(2)(2) ways to attempt.
As he has to attempt at least 1 question,the total number of ways in which he can attempt questions from 1st section is 2⁴-1.
Similarly for the 2nd section there are 2⁵-1 ways in which he can attempt and for the 3rd section there are 2⁶-1 ways.
The ways in which the attempts one or more questions in any section is independent of the number of ways in which he attempts one or more questions from the other sections.
Thus, total number of ways in which he can attempt questions in that paper
=> (2⁴-1)(2⁵-1)(2⁶-1) = 15*31*63 = 29295

Answer: B
Given there are 12 numbered chairs, such that chairs numbered 1 to 8 should be occupied.
_X_ , 2,3,4,5,6,7,_X_,9,10,11,12.
The various combination of chairs that ensures that no two men are sitting together are listed.
(1,3,5,__). The fourth chair can be 5,6,10,11 or 12, hence 5 ways.
(1,4,8,__), The fourth chair can be 6,10,11 or 12 hence 4 ways.
(1,5,8,__), the fourth chair can be 10,11 or 12 hence 3 ways.
(1,6,8,__), the fourth chair can be 10,11 or 12 hence 3 ways.
(1,8,10,12) is also one of the combinations.
Hence, 16 such combinations exist. In case of each these combinations we can make the four men inter arrange in 4! ways.
Hence, the required result = 16* 4! = 384

Answer: C
In order to reach (5,6) covering the shortest distance at the same time the man has to make 5 horizontal and 6 vertical steps.
The number of ways in which these steps can be taken is given by 11!/(5!*6!) = 462.