Permutation and combination question

[gaobo9109]

Homework Statement

A photographer is positioning 5 men and 4 women for a photo shoot. The men are positioned in the order from shortest on the left to tallest on the right. Find the number of ways the photographer can position them in a row. (All men are of different heights and are not necessarily standing together in a group)

Homework Equations

The Attempt at a Solution

Since the position of men are fixed, i will just consider the possible permutation of women.

Four women can either stand together as one group, or split into two, three or four groups.

Below are combination forl 5 possible cases
Grouped together: 6 x 4! = 360
Form two groups of 2: 4C2/2 x 6P2 = 90
Form two groups of 1 and 3: 4 x 6P2 = 120
Form three groups of 1,1 and 2: 4 x 3 x 6P3 = 1440
All separated: 6P4 = 360

Then I added up all the possible combinations. The answer I obtained is 2154. However, the answer provided is 3024. Can anyone tell me which case I fail to consider? Or is there anything wrong with my working? Thanks

 

[vela]

Homework Statement

A photographer is positioning 5 men and 4 women for a photo shoot. The men are positioned in the order from shortest on the left to tallest on the right. Find the number of ways the photographer can position them in a row. (All men are of different heights and are not necessarily standing together in a group)

Homework Equations

The Attempt at a Solution

Since the position of men are fixed, i will just consider the possible permutation of women.

Four women can either stand together as one group, or split into two, three or four groups.

Below are combination forl 5 possible cases
Grouped together: 6 x 4! = 360

6x24=144, not 360.

Form two groups of 2: 4C2/2 x 6P2 = 90

I'm too tired to figure out what you're doing, but I got 360 for this case.

Form two groups of 1 and 3: 4 x 6P2 = 120

And 720 for this case.

Form three groups of 1,1 and 2: 4 x 3 x 6P3 = 1440
All separated: 6P4 = 360

We agree on these cases.

Then I added up all the possible combinations. The answer I obtained is 2154. However, the answer provided is 3024. Can anyone tell me which case I fail to consider? Or is there anything wrong with my working? Thanks