how many ways can 4 married couples be arranged

sorry with all these questions, I'm jsut starting to learn this stuff, and it doesn't seem as im too good at it yet.

Okay another question, In how many ways can 4 married couples be arranged around a circular table if
c) each man sits beside his wife and mend and women alternate?
d) men and women alternate?

What would the difference be in these 2 questions, and how to do it if you have to make the men and women alternate?..

thanks again

If each man sits beside his wife the men and women would have designated seating so to speak.

If men and women only alternate, this means they don't have to sit beside their wive and they can sit beside any females.

You can try treating the men and women as one person if they have to alternate.

a): Husband and wife sit together and men and women alternate:

Let the couples be one person and arrange in (n-1)!=3!=6 ways.
The couple can be arranged in 2 ways. 6*2=12.
Also, you could seat a man, then there's one way to seat his wife to his right. Then one of the 3 remaining men, then his wife, and so on.
3*2*1=6. Pick the first man, only seat his wife to his left. Again, 3*2*1=6.
6+6=12 ways.




b): If men and women alternate, then suppose we seat a man first. We must then place a woman next out of 4. Then a man out of 3. Then a woman out of 3, and so on.

4*3*3*2*2*1*1=144 ways.

For part (b).
It is interesting to note that once one person is seated, the table is conceptually no longer circular. Therefore, the answer is indeed (3!)(4!)=144.

Hello, Anna!

If you're just starting, a "circular table" problem can be quite tricky.
. . But you already know that; you've done parts (a) and (b) already.
I'll baby-step through these for you . . .

In how many ways can 4 married couples be arranged around a circular table if
(c) each man sits beside his wife and mend and women alternate?
(d) men and women alternate?

Click to expand...

(c) Husbands and wives are adjacent and the genders alternate.

A husband can sit on the left or right of his wife.

And they would sit like this:
Duct-tape the couples together.
. . Their order will be either HW or WH ... \(\displaystyle 2\) choices.

Choose any couple and seat them in any pair of adjacent chairs.
Then the other three couples can be seated in \(\displaystyle 3!\) ways.

Therefore, there are: \(\displaystyle \,2\,\times\,3!\:=\:12\) ways.


(d) Men and women alternate.
Then there is basically one arrangement for the genders.
Choose any woman and seat her anywhere.

Then the other 3 women can occupy the W-seats in \(\displaystyle 3!\) ways.
And the 4 men can occupy the M-seats in \(\displaystyle 4!\) ways.

Therefore, there are: \(\displaystyle \,3!\,\times\,4!\:=\:144\) ways.

Baptize 'em: women= A,B,C,D and men= 1,2,3,4

As Soroban says, sit a woman down (make her A);
so seating arrangement can be shown this way:
A (1,2,3,4) (B,C,D) (1,2,3,4) (B,C,D) (1,2,3,4) (B,C,D) (1,2,3,4) A
(you can see that A has men on her right and left; and now forget it's a table!)

And the 144 arrangements will go this way:
1: A 1 B 2 C 3 D 4 A
2: A 2 B 1 C 3 D 4 A
and so on...

One must wonder why this question having been answered correctly two days ago, is once again answered. Was the first set of answers incorrect? Or was that set of answers judged inadequate to help the student even though the new answers are two days late?

Thanks for the explanations, and yea, I thought I got a and b...but no I didn't. Our teacher never told us that things that are circuar are different...it clarifies a lot now.