Question

How many ways are there for 3 men and 3 women stand in a line so that no two women stand next to each other?

Answer 1

so there must be a men between two women.
M F M F M F
F M F M F M

so you have 2 (3 * 3 * 2 * 2 * 1 * 1) <--

Answer 2

Hi sourwing! hmm so answer is 72?

Answer 3

hmm can't quite figure out, if there are no clashes, it is 6! but with this clash i not sure how to see it.

Answer 4

then take take and subtract the number of ways that you have at least 2 women standing next to each other

Answer 5

then the *that*...

Answer 6

omg I can't type XD

Answer 7

then *take that*...

Answer 8

@sourwing
How about
F M M F M F

Answer 9

How do I calculate the number of ways 2 woman together? Is it 4!?

Answer 10

O.O i missed that

Answer 11

Even if you count how many ways two women sit together, you have to consider three women sitting together, as well as double counting.

Answer 12

Ahh :(

Answer 13

So is it 4! * 2?

Answer 14

Seems there are four overall layouts:
M F M F M F
F M F M F M
F M M F M F
F M F M M F

Answer 15

let's see..
M F M F M F
F M F M F M
F M M F M F
F M F MM F

did i miss any thing? so 4 (3*3*2*2*1*1)

Answer 16

For each layout you permute the men 3! and then permute the women 3!

Answer 17

Looks like MWMWMW AND WMWMWM
Are the two configurations. The , the men can be reraanged 3! and so can the women 3! So? 2 x 3! x3! ? 72 If two women cannot stand together, cannot have WWW etc.

Answer 18

I get 4 * 3! * 3! = 144

Answer 19

Hmm can u explain how u get it?

Answer 20

I thought only MFMFMF and FMFMFM qualified.

Answer 21

The 3! is for choosing the order of men. The other 3! is for choosing the order of women.
The 4 is for choosing the gender allocation.

Answer 22

those and the other two ways we listed above

Answer 23

Is 144 correct?

Answer 24

I'm certain 144 is correct

Answer 25

I see I was wrong, as MM is ok. Just not FF. 144 it is.

Answer 26

Yea 144 is correct but trying to get things

Answer 27

@snowpolar so you get why it's 4 * 3! * 3!?

Answer 28

No

Answer 29

I won.

Answer 30

thumps up!! XD

Answer 31

Going to bed. Goodnight and goodluck.

Answer 32

@snowpolar so first, you get why it's 4 * something?

Answer 33

night night!!

Answer 34

No actually, I thought the solution is something like 6! - (5! * 2) but it isn't

Answer 35

it seems like you regard two woman as a unit and permutation their spot with the other 4 people. Well you can't do because you double counted some of them

FF FMMM

when you multiply by 2, you switch those two females that you move together and and still consider the combination different.

Answer 36

Hmm u mean?

Answer 37

well, when you do 5! you regard everyone is different from one another.
Say FFMFMM, and i switch places of the last two males, the combination is different, when in fact, it's the same.

Answer 38

Hmm so confusing!

Answer 39

it's actually confusing if you do it that way. The easiest way is the way I and Wio did it.

Answer 40

Can u explain your step in more detail?

Answer 41

well, you get these right?
M F M F M F
F M F M F M
F M M F M F
F M F MM F

Answer 42

Yep

Answer 43

for each row,

you have 3 ways for the fist F, 3 ways for the first male, then 2 ways the second F, 2 ways for the second M, then 1 way for the third F, and 1 way for the third M

Answer 44

for example:

for this row M F M F M F, you have 3*3*2*2*1*1
for this row F M M F M F, you have 3*3*2*2*1*1

but since you have 4 of them, just multiply 3*3*2*2*1*1 by 4

Answer 45

I see. But what if the number given is a big one then I had no choice but do it it mathematically without laying them all out.

Answer 46

you know, I thought the same. I myself still haven't have a solid understand of permutation and combination, especially when you make the conditions more complicated.

Answer 47

say you take the same problem that we did, and throw in more constraints, like, this female must be on the left of another female but on the right of this male.
You can always make things more complicated lol

Answer 48

Aha I see. Let me close this question.