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  =

\[{}^{10} C_4 \times {}^6 C_6 \times 8! \times 8! = {10}_{C_4} \times \left( 8! \right)^2\]