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 be seated?

Answer 1

A tea party is arranged for 16 people along two sides of a long table with 8 chairs on each side.
4 people wish to sit on side $A$ (say) and two on side $B$ (say).
Now, 10 people are left, out of which 4 people can be selected for side $A$ in ¹⁰C₄ ways.
And, from the remaining people, 6 people can be selected for side B in ⁶C₆ ways.
∴ Number of selections =${}^{10}{C}_4\times {}^6{C}_6$
Now, 8 people on each side can be arranged in $8!$ ways.
∴ Total number ways in which the people can be seated =${}^{10}{C}_4\times {}^6{C}_6\times 8!\times 8!={10}_{C_4}\times {\left(8!\right)}^2$