Question

In how many different ways a grandfather along with two of his grandsons and four granddaughters can be seated in a line for a photograph so that he is always in the middle and the two grandsons are never adjacent to each other.

Answer 1

Hint: In order to solve such a question, use the concept of permutation and factorial. Find out the total number of ways of seating arrangement and then subtract the number of ways possible such that boys are adjacent from the total number of ways.

Complete step-by-step answer:
Total number of family members = 7
Total no of seats = 7
Since the grandfather sits in the middle so no of left seat = 6
So total number of arrangements possible for 6 seats $ = 6!$
Now in order to find the number of arrangements in which boys are not at adjacent positions.
Let us first find out the number of ways in which boys are sitting adjacent.
In order to find this, we will consider both the boys as one unit. So we are left with 5 units for arrangement.
Also the boys can sit together in 2 ways, any boy on either side.
$ \Rightarrow $ Number of ways boys are sitting together $ = 5!2!$
There may be cases included above when both the boys are sitting on either side of the middle seat. So, we have to remove such conditions.
$ \Rightarrow $ Number of ways boys are sitting together but in the middle two spot $ = 4!2!$
So, the number of ways boys can sit together $ = 5!2! - 4!2!$
Now in order to find the number of required arrangements. We will subtract the number of unwanted arrangements (arrangements where boys are sitting together) from the total number of arrangements.
$
   = 6! - \left( {5!2! - 4!2!} \right) \\
   = 6! - 5!2! + 4!2! \\
   = 720 - \left( {120 \times 2} \right) + \left( {24 \times 2} \right) \\
   = 720 - 240 + 48 \\
   = 528 \\
 $
Hence, the total number of ways the grandfather along with his descendants can be seated is 528.

Note: For solving such questions related to number of ways or number of arrangements, formulas of factorial and permutation are very important for easy solutions. Also as in the case above the number of required arrangements cannot be found out directly. In such cases try to find out the total number of arrangements and further subtract unwanted numbers of arrangements.