# How many ways are there for four men and five women ..

1. Feb 13, 2019 at 11:08 AM

### r0bHadz

1. The problem statement, all variables and given/known data
How many ways are there for four men and five women to stand in a line so that

All men stand together?

All women stand together?

2. Relevant equations

3. The attempt at a solution
For all men stand together, you can group the 4 men as one token, then there are (1+5)! ways the men and women can stand in a line, but the four men can be arranged in 4! ways so the answer would be (1+5)!(4!) = (6!)(4!)

similar for women
let 5 women = one token, then you have (1+4)! but the 5 women can be arranged 5! ways so you have (5!)(5!) as the answer

is there anything I am missing?

2. Feb 13, 2019 at 12:59 PM

### Staff: Mentor

Moderator's Note: Thread moved to pre-calculus math homework forum.

3. Feb 13, 2019 at 7:10 PM

### WWGD

Men can go in positions 1-5, 2-6, etc.

4. Feb 14, 2019 at 1:17 AM

### r0bHadz

So I am taking the 4 men as one line. There are 4! possible combinations for the men. Then, There are 5 women, and the group of four men which I'm considering as 1. There are 6! possible arrangements here.

Wouldn't "men can go in positions 1-5,2-6, etc" be under the (6!)?

5. Feb 14, 2019 at 2:33 AM

### PeroK

That's right. You could have tested your approach with smaller numbers, perhaps 2 and 3.

6. Feb 14, 2019 at 2:38 AM

### r0bHadz

Sorry are both answers correct? WWGD's post has me a little paranoid lol

7. Feb 14, 2019 at 2:49 AM

### PeroK

Yes, the answers are correct.

That's another way to do it. For the men as a group, they have 6 possible positions (with the first man in position 1-2-3-4-5 or 6). Then it's $6 \times 4! \times 5!$, which is the same as you got.

8. Feb 14, 2019 at 5:51 PM

### WWGD

Don't worry, my reply agrees with PeroK's and yours.

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