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Question Number 206681 by cortano21 last updated on 22/Apr/24
   s
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Commented by A5T last updated on 22/Apr/24
This question is similar to Q202257, does it  also mean every wife cannot come before her  husband?
$${This}\:{question}\:{is}\:{similar}\:{to}\:{Q}\mathrm{202257},\:{does}\:{it} \\ $$$${also}\:{mean}\:{every}\:{wife}\:{cannot}\:{come}\:{before}\:{her} \\ $$$${husband}? \\ $$
Commented by mr W last updated on 22/Apr/24
how can a wife go through her  husband? please illustrate what  you mean.
$${how}\:{can}\:{a}\:{wife}\:{go}\:{through}\:{her} \\ $$$${husband}?\:{please}\:{illustrate}\:{what} \\ $$$${you}\:{mean}. \\ $$
Commented by cortano21 last updated on 22/Apr/24
    _(1 ) ,  ∨2
$$\:\:\cancel{\underline{\underbrace{\:}}} _{\mathrm{1}\:} ,\:\:\underline{\vee}\mathrm{2} \\ $$
Commented by mr W last updated on 22/Apr/24
you mean a wife may not stand   before her husband in the queue.
$${you}\:{mean}\:{a}\:{wife}\:{may}\:{not}\:{stand}\: \\ $$$${before}\:{her}\:{husband}\:{in}\:{the}\:{queue}. \\ $$
Commented by cortano21 last updated on 22/Apr/24
 ξ
$$\:\cancel{\xi} \\ $$
Commented by A5T last updated on 22/Apr/24
2520 ?
$$\mathrm{2520}\:? \\ $$
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−1) +^n P_2 ×(2n−2)(2n−3)T_(n−2)   ⇒T_n =nT_(n−1) +2n(n−1)^2 (2n−3)T_(n−2)   T_1 =1,T_2 =6⇒T_3 =3×6+2×3×4×3×1=90  ⇒T_4 =4×90+2×4×9×5×6=2520.
$${Recursive}\:{formula}\:{I}\:{got}: \\ $$$${T}_{{n}} ={number}\:{of}\:{possible}\:{arrangements}\:{for}\:{n}\:{couples} \\ $$$${T}_{{n}} ={nT}_{{n}−\mathrm{1}} +^{{n}} {P}_{\mathrm{2}} ×\left(\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{3}\right){T}_{{n}−\mathrm{2}} \\ $$$$\Rightarrow{T}_{{n}} ={nT}_{{n}−\mathrm{1}} +\mathrm{2}{n}\left({n}−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2}{n}−\mathrm{3}\right){T}_{{n}−\mathrm{2}} \\ $$$${T}_{\mathrm{1}} =\mathrm{1},{T}_{\mathrm{2}} =\mathrm{6}\Rightarrow{T}_{\mathrm{3}} =\mathrm{3}×\mathrm{6}+\mathrm{2}×\mathrm{3}×\mathrm{4}×\mathrm{3}×\mathrm{1}=\mathrm{90} \\ $$$$\Rightarrow{T}_{\mathrm{4}} =\mathrm{4}×\mathrm{90}+\mathrm{2}×\mathrm{4}×\mathrm{9}×\mathrm{5}×\mathrm{6}=\mathrm{2520}. \\ $$
Commented by cortano21 last updated on 22/Apr/24
 ζ
$$\:\underbrace{\zeta} \\ $$$$ \\ $$
Commented by A5T last updated on 22/Apr/24
number of possible arrangements for n−1 couples
$${number}\:{of}\:{possible}\:{arrangements}\:{for}\:{n}−\mathrm{1}\:{couples} \\ $$
Answered by A5T last updated on 22/Apr/24
T_n =(((2n)!)/2^n )⇒T_4 =((8!)/(16))=2520
$${T}_{{n}} =\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} }\Rightarrow{T}_{\mathrm{4}} =\frac{\mathrm{8}!}{\mathrm{16}}=\mathrm{2520} \\ $$
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  (2n)! possibilities. when we look at  the couple no. 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. 1 the number   of ways that the wife is behind the  husband is (((2n)!)/2). similarly amony  these possibilities, for the couple  no. 2 the number of ways that the  wife is behind the husband is (((2n)!)/(2×2)).  in this way we get for all couples the  number of ways that the wife of   each couple is behind her husband is  (((2n)!)/(2×2×2×...×2))=(((2n)!)/2^n ) ✓  for 4 couples the answer is ((8!)/2^4 )=2520
$${let}'{s}\:{say}\:{there}\:{are}\:{n}\:{couples}.\:{to} \\ $$$${arrange}\:{them}\:{in}\:{a}\:{queue},\:{there}\:{are} \\ $$$$\left(\mathrm{2}{n}\right)!\:{possibilities}.\:{when}\:{we}\:{look}\:{at} \\ $$$${the}\:{couple}\:{no}.\:\mathrm{1},\:{due}\:{to}\:{symmetry} \\ $$$${there}\:{are}\:{exactly}\:{so}\:{many} \\ $$$${possibilities}\:{that}\:{the}\:{wife}\:{is}\:{behind} \\ $$$${the}\:{husband}\:{as}\:{the}\:{possibilities} \\ $$$${that}\:{the}\:{wife}\:{is}\:{before}\:{the}\:{husband}, \\ $$$${so}\:{for}\:{the}\:{couple}\:{no}.\:\mathrm{1}\:{the}\:{number}\: \\ $$$${of}\:{ways}\:{that}\:{the}\:{wife}\:{is}\:{behind}\:{the} \\ $$$${husband}\:{is}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}}.\:{similarly}\:{amony} \\ $$$${these}\:{possibilities},\:{for}\:{the}\:{couple} \\ $$$${no}.\:\mathrm{2}\:{the}\:{number}\:{of}\:{ways}\:{that}\:{the} \\ $$$${wife}\:{is}\:{behind}\:{the}\:{husband}\:{is}\:\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}×\mathrm{2}}. \\ $$$${in}\:{this}\:{way}\:{we}\:{get}\:{for}\:{all}\:{couples}\:{the} \\ $$$${number}\:{of}\:{ways}\:{that}\:{the}\:{wife}\:{of}\: \\ $$$${each}\:{couple}\:{is}\:{behind}\:{her}\:{husband}\:{is} \\ $$$$\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}×\mathrm{2}×\mathrm{2}×…×\mathrm{2}}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} }\:\checkmark \\ $$$${for}\:\mathrm{4}\:{couples}\:{the}\:{answer}\:{is}\:\frac{\mathrm{8}!}{\mathrm{2}^{\mathrm{4}} }=\mathrm{2520} \\ $$

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