Question

Six couples are standing in a single-file line at a movie theater in such a way that no partner is separated. How many such lineups are possible?

Answer 1

2!6!

Answer 2

Every couple can be arranged in 2! ways = 2 ways
Also, the 6 couples can be arranged in 6! ways
Hence, total number of ways = 6! x 6(2!) = 46080 ways

Answer 3

why 6(2!) ??

Answer 4

no it's not those....and i would also appreciate some explanation not just direct answers...

Answer 5

Since every couple can be arranged in 2! ways, the 6 couples can be arranged in 2! x 2! x 2! x 2! x 2! x 2! = 6(2!)

Answer 6

that's (2!)^6 not 6(2!)....

Answer 7

Since every couple can be arranged in 2! ways, the 6 couples can be arranged in 2! x 2! x 2! x 2! x 2! x 2! = 6(2!) among themselves

Answer 8

and im still asking for an explanation....

Answer 9

6!(2!)^6 is the right solution though

Answer 10

Every couple can be arranged in 2! ways = 2 ways
Also, the 6 couples can be arranged in 6! ways
Hence, total number of ways = 6! x (2!) ways

Answer 11

That is true

Answer 12

O.o

Answer 13

now...why 6!(2!)^6? i don't get it

Answer 14

@KKJ
u first wrote
"the 6 couples can be arranged in 6! ways"
then wrote
" the 6 couples can be arranged in 2! x 2! x 2! x 2! x 2! x 2! = 6(2!) among themselves"

Answer 15

let me try and see if i get it... 2! because the couples cannot be separated so it's 2!

right?

Answer 16

So, it is 6! x (2!)^6 = 46080 ways

Answer 17

no it's 6!(2!)^6

Answer 18

It was a typographical error

Answer 19

anyway...am i right with my question?

Answer 20

yes. lg
and u probably also understood 6!
now only question is why ^6 ??

Answer 21

2! came from the fact that the couples cannot be separated that's why 2!

??

Answer 22

Remember, the positions of the couples could be altered but they could still stand together

Answer 23

@hartnn you're a lucky guy...you got the medals but the right person didn't

Answer 24

anyway..back to my question...

Answer 25

Also each of the couples are arranged in 2! ways

Answer 26

you pack the couple. you see 6 packs. 6 packs can be arranged in 6! ways.

Answer 27

...im just asking a yes/no question...

Answer 28

lol, i didn't ask for...
@KKJ position of couples is covered in 6! ways
why do i raise 2! to power of 6 ??

Answer 29

@hartnn 6!(2!)^6 is the right solution

Answer 30

yes, u told me , now my only doubt is why ^6

Answer 31

2! 2! 2! 2! 2! 2! 2! <--i assume that's because there are 6 couples

Answer 32

then 6! is because...what?

Answer 33

for each of the couples could stand anywhere in the line up and still be together

Answer 34

as @ganeshie8 explained, treat a couple as 1, then they can be arranged among themselves in 6! ways.

Answer 35

6! is for all of them to be in a straight line

Answer 36

ahh yes that makes sense

Answer 37

hartnn's medals really confuse me....

Answer 38

"for each of the couples could stand anywhere in the line up and still be together"
just means 6! ways of arranging, not 2!^6

Answer 39

so 2! 2! 2! 2! 2! 2! is for the arrangement of the couples in EVERY slot?

Answer 40

I think it was 2*6! but now I think it is 6!*(2^6)

Answer 41

So if the first couple stand in the first position to make the line up
or they stand in the second position for the line up
. . . and so on

Answer 42

still no @sauravshakya ....

Answer 43

Because each coulpe can be arranged in two ways

Answer 44

huh im still with 6! 2!

Answer 45

ahh yes that makes sense

Answer 46

nope it's 6! (2!)^6 @ganeshie8 (for the nth time)

Answer 47

http://openstudy.com/updates/505ba3dee4b03290a415b1cc

just so you know im not agreeing with KKJ just now

Answer 48

Questions of this nature need serious analysis

Answer 49

You should imagine the arrangement in the real sense

Answer 50

indeed. these things are tricky

Answer 51

If not, you will just do anything and think it's true

Answer 52

somehow it resembles the solution for this kind of question "Judy has three sets of classics in literature, each set having four volumes. In how many ways can she put them in a bookshelf so that books of each set are not separated?"

Answer 53

indeed. they are very similar...

Answer 54

anyway i have to go now. thanks for explaining @KKJ ..nice defense too

Answer 55

Actually, the one you just gave is made up of distinguishable items

Answer 56

Thanks Igbasallote

Answer 57

ahh i see hw its 6! 2!^6, we arrange 6 packs in 6!, and we can open any of the 6 packs and arrange them in 2 ways. 2! * 2! * 2! * 2! * 2! * 2! = 2!^6

good one @lgbasallote :)

Answer 58

agreed with @ganeshie8

Answer 59

those who gave me medals can take it away...
i generally do this kind of problems by taking the simplest case.So lets take 2 couples AB and CD, so there will be 8 lines possible, 4 as shown below and other 4 by exchanging A with C and B with D
A B A B
B A B A
C C D D
D D C C
hence here ways =8
so for 6 couples, its indeed 6! * 2!^6

Answer 60

finally @hartnn is 99

Answer 61

yup, in 47 days......thanks.