Question

$7$ boys and $8$ girls have to sit in a row on $15$ chairs numbered from $1$ to $15$ then?

• A. Number of ways boys and girls sit alternately is $8!7!$
• B. Number of ways boys and girls sit alternately is $2(8!7!)$
• C. The number of ways in which first and fifteenth chair are occupied by boys and between any two boys an even number of girls sit is ${}^9{C}_4$ $8!7!$
• D. The number of ways in which first and last seat are occupied by boys and between any two boys an even number of girls sit is $(2{}^9{C}_{4}8!7!)$.

Answer 1

The correct option is C The number of ways in which first and fifteenth chair are occupied by boys and between any two boys an even number of girls sit is ${}^9{C}_4$ $8!7!$

$\begin{array}{c}G-denotesGirls\\ X-denotesboys\\ \Rightarrow G\times G\times G\times G\times G\times G\times G\times G\\ \Rightarrow NumberofwaysofGirlsis8!alternate\\ \Rightarrow Numberofwaysofboysis7!\\ afterlate\\ \Rightarrow Numberofboysandgirlssitalternate=8!7!\\ Whenfirstandfifteenchairarefixedand\\ firstchair\times G\times G\times G\times G\times G\times G\times G\times fifteenchairbetweenanytwoboys\\ \underset{–}{9C_{4}8!7!}Ans.\end{array}$