Question Number 206681 by cortano21 last updated on 22/Apr/24
$$\:\:\:\cancel{{s}} \\ $$$$ \\ $$
Commented by A5T last updated on 22/Apr/24
$${This}\:{question}\:{is}\:{similar}\:{to}\:{Q}\mathrm{202257},\:{does}\:{it} \\ $$$${also}\:{mean}\:{every}\:{wife}\:{cannot}\:{come}\:{before}\:{her} \\ $$$${husband}? \\ $$
Commented by mr W last updated on 22/Apr/24
$${how}\:{can}\:{a}\:{wife}\:{go}\:{through}\:{her} \\ $$$${husband}?\:{please}\:{illustrate}\:{what} \\ $$$${you}\:{mean}. \\ $$
Commented by cortano21 last updated on 22/Apr/24
$$\:\:\cancel{\underline{\underbrace{\:}}} _{\mathrm{1}\:} ,\:\:\underline{\vee}\mathrm{2} \\ $$
Commented by mr W last updated on 22/Apr/24
$${you}\:{mean}\:{a}\:{wife}\:{may}\:{not}\:{stand}\: \\ $$$${before}\:{her}\:{husband}\:{in}\:{the}\:{queue}. \\ $$
Commented by cortano21 last updated on 22/Apr/24
$$\:\cancel{\xi} \\ $$
Commented by A5T last updated on 22/Apr/24
$$\mathrm{2520}\:? \\ $$
Answered by A5T last updated on 22/Apr/24
$${Recursive}\:{formula}\:{I}\:{got}: \\ $$$${T}_{{n}} ={number}\:{of}\:{possible}\:{arrangements}\:{for}\:{n}\:{couples} \\ $$$${T}_{{n}} ={nT}_{{n}−\mathrm{1}} +^{{n}} {P}_{\mathrm{2}} ×\left(\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{3}\right){T}_{{n}−\mathrm{2}} \\ $$$$\Rightarrow{T}_{{n}} ={nT}_{{n}−\mathrm{1}} +\mathrm{2}{n}\left({n}−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2}{n}−\mathrm{3}\right){T}_{{n}−\mathrm{2}} \\ $$$${T}_{\mathrm{1}} =\mathrm{1},{T}_{\mathrm{2}} =\mathrm{6}\Rightarrow{T}_{\mathrm{3}} =\mathrm{3}×\mathrm{6}+\mathrm{2}×\mathrm{3}×\mathrm{4}×\mathrm{3}×\mathrm{1}=\mathrm{90} \\ $$$$\Rightarrow{T}_{\mathrm{4}} =\mathrm{4}×\mathrm{90}+\mathrm{2}×\mathrm{4}×\mathrm{9}×\mathrm{5}×\mathrm{6}=\mathrm{2520}. \\ $$
Commented by cortano21 last updated on 22/Apr/24
$$\:\underbrace{\zeta} \\ $$$$ \\ $$
Commented by A5T last updated on 22/Apr/24
$${number}\:{of}\:{possible}\:{arrangements}\:{for}\:{n}−\mathrm{1}\:{couples} \\ $$
Answered by A5T last updated on 22/Apr/24
$${T}_{{n}} =\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} }\Rightarrow{T}_{\mathrm{4}} =\frac{\mathrm{8}!}{\mathrm{16}}=\mathrm{2520} \\ $$
Answered by mr W last updated on 23/Apr/24
$${let}'{s}\:{say}\:{there}\:{are}\:{n}\:{couples}.\:{to} \\ $$$${arrange}\:{them}\:{in}\:{a}\:{queue},\:{there}\:{are} \\ $$$$\left(\mathrm{2}{n}\right)!\:{possibilities}.\:{when}\:{we}\:{look}\:{at} \\ $$$${the}\:{couple}\:{no}.\:\mathrm{1},\:{due}\:{to}\:{symmetry} \\ $$$${there}\:{are}\:{exactly}\:{so}\:{many} \\ $$$${possibilities}\:{that}\:{the}\:{wife}\:{is}\:{behind} \\ $$$${the}\:{husband}\:{as}\:{the}\:{possibilities} \\ $$$${that}\:{the}\:{wife}\:{is}\:{before}\:{the}\:{husband}, \\ $$$${so}\:{for}\:{the}\:{couple}\:{no}.\:\mathrm{1}\:{the}\:{number}\: \\ $$$${of}\:{ways}\:{that}\:{the}\:{wife}\:{is}\:{behind}\:{the} \\ $$$${husband}\:{is}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}}.\:{similarly}\:{amony} \\ $$$${these}\:{possibilities},\:{for}\:{the}\:{couple} \\ $$$${no}.\:\mathrm{2}\:{the}\:{number}\:{of}\:{ways}\:{that}\:{the} \\ $$$${wife}\:{is}\:{behind}\:{the}\:{husband}\:{is}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}×\mathrm{2}}. \\ $$$${in}\:{this}\:{way}\:{we}\:{get}\:{for}\:{all}\:{couples}\:{the} \\ $$$${number}\:{of}\:{ways}\:{that}\:{the}\:{wife}\:{of}\: \\ $$$${each}\:{couple}\:{is}\:{behind}\:{her}\:{husband}\:{is} \\ $$$$\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}×\mathrm{2}×\mathrm{2}×…×\mathrm{2}}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} }\:\checkmark \\ $$$${for}\:\mathrm{4}\:{couples}\:{the}\:{answer}\:{is}\:\frac{\mathrm{8}!}{\mathrm{2}^{\mathrm{4}} }=\mathrm{2520} \\ $$