Question

Find the number of ways in which 6 boys and 5 girls can dine on a round table, if 2 girls want to sit together?

Answer 1

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{two}\mathrm{girls}\mathrm{which}\mathrm{want}\mathrm{to}\mathrm{sit}\mathrm{together}\mathrm{as}\mathrm{one}. \\ \mathrm{So},\mathrm{now}\mathrm{total}\mathrm{no}.\mathrm{of}\mathrm{boys}\mathrm{and}\mathrm{girls}\mathrm{are}10. \\ \mathrm{They}\mathrm{can}\mathrm{sit}\mathrm{in}\left(10-1\right)!=9! \\ \mathrm{And}\mathrm{that}2\mathrm{girl}\mathrm{also}\mathrm{can}\mathrm{be}\mathrm{arrange}\mathrm{themselves}\mathrm{in}2!\mathrm{ways}=2 \\ \mathrm{So},\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{ways}=9!\times 2 \\ =362880\times 2 \\ =725760\mathrm{Ways}. \end{gathered}$$