Question Number 218299 by mr W last updated on 05/Apr/25
$$\mathrm{10}\:{couples}\:{are}\:{invited}\:{to}\:{a}\:{dinner}\:{and} \\ $$$${should}\:{be}\:{seated}\:{at}\:{a}\:{round}\:{table}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{host}\:{do} \\ $$$${this}, \\ $$$$\left.\mathrm{1}\right)\:{generally} \\ $$$$\left.\mathrm{2}\right)\:{if}\:{two}\:{men}\:{should}\:{not}\:{sit}\:{next} \\ $$$$\:\:\:\:\:{to}\:{each}\:{other} \\ $$$$\left.\mathrm{3}\left.\right)\:{as}\:\mathrm{2}\right),\:{but}\:{two}\:{speicial}\:{couples}\: \\ $$$$\:\:\:\:\:{should}\:{not}\:{sit}\:{next}\:{to}\:{each}\:{other} \\ $$
Commented by Tawa11 last updated on 05/Apr/25
$$\mathrm{I}\:\mathrm{just}\:\mathrm{try}\:\mathrm{sir}. \\ $$$$\left(\mathrm{1}\right)\:\:\left(\mathrm{20}\:−\:\mathrm{1}\right)!\:\:=\:\mathrm{19}! \\ $$$$\left(\mathrm{2}\right)\:\:\:\mathrm{9}!\:×\:\mathrm{10}! \\ $$$$\left(\mathrm{3}\right)\:\:\:\mathrm{9}!\:×\:\mathrm{10}!\:−\:\mathrm{2}\:×\:\mathrm{8}!\:×\:\mathrm{10}! \\ $$
Commented by mr W last updated on 05/Apr/25
$${please}\:{explain}\:{your}\:{solution}\:{for}\:\left(\mathrm{3}\right). \\ $$$$ \\ $$$${say}\:{H}_{{i}} \:{are}\:{the}\:{husbands}\:{and}\:{W}_{{i}} \:{are} \\ $$$${the}\:{wives}.\:\left({i}=\mathrm{1},\mathrm{2},…,\mathrm{10}\right) \\ $$$${H}_{\mathrm{1}} \:{and}\:{W}_{\mathrm{1}} \:{are}\:{the}\:{couple}\:\mathrm{1},\: \\ $$$${H}_{\mathrm{2}} \:{and}\:{W}_{\mathrm{2}} \:{are}\:{the}\:{couple}\:\mathrm{2}.\:{It}\:{is} \\ $$$${meant}\:{that}\:{H}_{\mathrm{1}} \:{may}\:{not}\:{sit}\:{next}\:{to} \\ $$$${W}_{\mathrm{2}} \:{and}\:{H}_{\mathrm{2}} \:{may}\:{not}\:{sit}\:{next}\:{to}\:{W}_{\mathrm{1}} . \\ $$$${but}\:{certainly}\:{H}_{\mathrm{1}} \:{may}\:{sit}\:{next}\:{to}\:{W}_{\mathrm{1}} \\ $$$${and}\:{H}_{\mathrm{2}} \:{may}\:{sit}\:{next}\:{to}\:{W}_{\mathrm{2}} . \\ $$
Commented by mehdee7396 last updated on 05/Apr/25
$$ \\ $$$$\mathrm{9}!×\mathrm{8}!\: \\ $$