95. A family comprised of an old man, 6 adults and 4 children is to be seated is a row with
the condition that the children would occupy both the ends and never occupy either side
of the old man. How many sitting arrangements are possible?
need explanation
Answers
Given:
A family comprised of an old man, 6 adults and 4 children is to be seated is a row with the condition that the children would occupy both the ends and never occupy either side of the old man.
To find:
How many sitting arrangements are possible?
Solution:
A family comprised of an old man, 6 adults and 4 children.
1 + 6 + 4 = 11 persons and similarly 11 seats.
Children would occupy both the ends and never occupy either side of the old man.
There are 4C2 ways to select 2 children.
These two children can be seated in the side seats in 2! ways.
The old man can occupy any of the 7 middle seats(7 ways).
An old man cannot occupy 2nd and 10th seat as they are by the side of seats of children.
3 seats are occupied and 8 seats are remaining.
In these 8 seats, 2 seats will be near to the old man in which children cannot seat.
Thus 6 seats are remaining in which 2 children can be seated.
This is given by 6P2 ways.
5 persons are seated now.
Remaining 6 persons can be arranged in the remaining 6 seats in 6! ways.
Therefore, the required number of ways
= 4C2 × 2! × 7 × 6P2 × 6!
= 6 × 2 × 7 × 30 × 720
= 1814400
Therefore, 1814400 sitting arrangements are possible.
(a) 4!×5!×7!
CHILDREN (4)
First we will arrange children as they have to occupy only end seats (1st,2nd,10th&11th) & they are 4 in number so 4!
OLD MAN (1)
then secondly we will arrange old Man as 4 seats (1st,2nd,10th& 11th) are already occupied & he cannot sit on either side of children which means he cannot sit on seats number 3rd & 9th so now he only have 5 seats left to choose (4th,5th,6th,7th&8th) i.e. 5!
ADULTS (7)
lastly we will see Adults As seats (1st,2nd,10th&11th) are already occupied by children. They have 7 seats left (3rd,4th,5th,6th,7th,8th&9th) to choose so 7!
Important Part of this question is that In Adult case old Man will not be considered fixed (like children) because adults & old man may permute between seats (4th,5th,6th&7th)
I hope you will Understand my Answer
