Question

$18$ guests have to be seated half on each side of a long table. $4$ particular guests desire to sit on one particular side and $3$ others on the other side. Determine the number of ways in which the sitting arrangements can be made.

• A. $(9!{)}^2$
• B. ${}^{11}{C}_4(9!{)}^2$
• C. ${}^{11}{C}_3(9!{)}^2$
• D. ${}^{11}{C}_5(9!{)}^2$

Answer 1

Answer: (D)

${}^{11}{C}_5(9!{)}^2$

Let the two sides be $A$ and $B$.

Assume that four particular guests wish to sit on side $A$.

Four guests who wish to sit on side $A$ can be accommodated on nine chairs in ${}^9{P}_4$ ways,

and three guests who wish to sit on on side $B$ can be accommodated on nine chairs in ${}^9{P}_3$ ways.

Now, the remaining guests are left who can sit on $11$ chairs on both the sides of the table in $(11!)$ ways.

Hence, the total number of ways in which $18$ persons can be seated,

${=}^9{P}_4{\times }^9{P}_3\times (11!)$

$=$${}^{11}{C}_5(9!{)}^2$