Question

A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they are seated?

Answer 1

Given that,

16 persons need to be seated along two sides of a long table with 8 persons on each side. 

It is also told that 4 persons sit on a particular side and 2 on the other side. 

We need to choose 4 members to sit on one side from the remaining 10 members as the 6 members are already fixed about their sittings and arrange 8 members on both sides accordingly. 

Let us first find the no. of ways to choose 4 members and assume it to be N₁. 

⇒ N₁ = Selecting 4 members from remaining 7 person members 

⇒ N₁ = ¹⁰C₄ 

We know that,

ⁿC_r = \(\frac{n!}{(n-r)!r!}\)

And also,

n! = (n)(n – 1)......2.1

⇒ N₁ = 210 

Now,

We need to arrange the chosen 8 members. 

Since 1 person differs from other. 

The arrangement is similar to that of arranging n people in n places which are n! Ways to arrange. 

So, 

The persons can be arranged in 8! Ways. 

This will be the same for both the tables. 

Let us assume the total possible arrangements be N. 

⇒ N = N₁ × 8! × 8! 

⇒ N = 210 × 8! × 8! 

∴ The total no. of ways of seating arrangements can be done 210 × 8! × 8!.