There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats so that neither little boy sits to the side of either little girl. In how many different ways can these six people be seated?
OPTIONS
1) 48
2) 128
3) 192
4) 176

- anonymous

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- mattfeury

I am inclined to think:
\[2!^3 * 3! = 48 \]

- anonymous

you have BBGGAA

- anonymous

can we think in linear way??

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## More answers

- mattfeury

what do you mean?

- anonymous

how did you get that answer

- anonymous

or kindely explain how u did?

- mattfeury

if you a have of couple (2 people), you can arrange them in 2! ways.
2! = 2 * 1 = 2
if you have three couples in 3 rows, then they can be sat = \[2! * 2! * 2!\]

- mattfeury

but! it can be:
couple1 couple2 couple3
couple3 couple2 couple1
etc...
so we can arrange the 3 couples in 3! ways.

- mattfeury

so basically we have 6 seats. but first we simply that to three rows.
we have three rows. a couple per row. there are 3! ways to arrange this. in each row, each couple can be arranged 2! ways. Since this is the case for each of the three rows, it is 2! * 2! * 2!
multiplying all these possibilities together:
2! * 2! * 2! * 3! = 48

- anonymous

you're treating the two boys as a couple?

- mattfeury

ah yeah sorry. i was thinking of a similar problem with 3 couples. it is the same principle though. 3 sets of 2 individuals (in this case: boys, girls, adults).

- anonymous

but you can rearrange the couples

- mattfeury

yes. you can arrange the couples between the three rows. thus the 3!

- anonymous

we have two rows
_ _ _
_ _ _

- mattfeury

ah i am so wrong. you are right.

- anonymous

yes thats what m thinking .. how to solve its definitely not circular and I doubt on Linear permutation

- anonymous

thats why i got confused about your answer

- mattfeury

heh sorry. i misread the question :(. that's what i get for skimming.

- anonymous

i will forgive you if you help me with my question, brb

- anonymous

but go ahead and try to answer

- anonymous

so you solved the question, 3 rows, boy cant sit next to girl?

- anonymous

so i guess is it correct??
B1 B2 A1 ----6 WAYS
G1 G2 A2 ----6 WAYS ie, TOTAL =36 ways

- mattfeury

there is another set of cases:
B1 A1 G1
B2 A2 G2

- mattfeury

B1 A1 G1 = 2 ways to arrange
B2 A2 G2 = 2 ways to arrange

- mattfeury

better:
B1 A1 G1 = 4 * 3 to arrange (you have a set of {b1, b2, g1, g2} and you have 4 and 3 options for the end)
B2 A2 G2 = 2 * 1 ways to arrange

- mattfeury

12 * 2 * 2 (another 2 to choose between the A's) = 48 for that case.

- mattfeury

times another 2 to account for being able to switch the rows: r1, r2 or r2, r1

- mattfeury

= 96 for that case.
you'll have to + this to the possibilities for the other case

- mattfeury

i may be wrong though. i can't seem to figure out the other case to where it adds to an answer :(
96 + (2 * 2 * 2 * 2) = 112?

- anonymous

I am also confused dear :)

- mattfeury

128 maybe!

- anonymous

can it be like?
B A G
B A G
G A B
G A B
B B A
G G A
G G A
B B A
A B B
A G G
A G G
A B B
Each case can be arranged in 8 ways , i.e. Boys - 2! , Gals - 2! , adults - 2!
Hence 6 * (2*2*2) = 48
Option 1

- anonymous

hi cantorset .. u ther??

- nowhereman

The only possible cases are, if the girls sit on different rows, that the parents sit in the middle seats and the boys on the other sides. If the girls sit on the same row the third seat must be occupied by a parent and there is no further restriction on the other row. So you get for the number of constellations: In the first case 4*2!*2!*2! = 32 and in the second case 2*3*3*2!*2!*2! = 144, that means there are 176 possibilties.

- nowhereman

need more explanation?

- anonymous

yes i do, i got the same answer but i didnt get your 2x3x3x2! , etc

- nowhereman

2 because the girls can either sit in the first or in the second row
3 because you can choose any seat in that row for the parent
3 because you must choose one seat in the other row where the second parent sits
2!*2!*2! always because it doesn't matter if you interchange the 2 girls / boys / parents with each other.

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