SOLUTION: Four men and eight women are to line up for a picture with the four men in the middle. How many ways can this be done? (Assume there are four women on each side of the group of men

Question 842435: Four men and eight women are to line up for a picture with the four men in the middle. How many ways can this be done? (Assume there are four women on each side of the group of men)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the 4 men in the middle can be arranged 4! ways.
the 8 women all in a row can be arranged in 8! ways.
the total number of ways will be 4! * 8! = 967,680 ways
you take the 8 women in a row and split them in half and stick the 4 men in the middle.
4! is equal to 24
8! is equal to 40320.
for each of the 40320 arrangements of the women, you can stick 24 arrangements of the men in the middle.
this makes the total number of arrangements equal to 24 * 40320 which is equal to 967680.

let's assume smaller numbers so you can see what's happening.
assume 2 men and 4 women.
the men can be arranged in 2! ways which is equal to 2.
the women can be arranged in 4! ways which is equal to 24.
the total ways they can be arranged is 2 * 24 = 48.
let letters represent the men and let number represent the women.
the letters will be a and b.
the numbers will be 1 and 2 and 3 and 4.
the men can be arranged as:
ab
ba
the women can be arranged as:

1234
1243
1324
1342
1423
1432
2134
2143
2314
2341
2413
2431
3124
3142
3214
3241
3412
3421
4123
4132
4213
4231
4312
4321

all you do now is split the 4 women down the middle and stick the 2 men in between.
each of the 24 arrangements for the women will contain 2 arrangements for the men, so you will wind up with 2 * 24 = 48 total arrangements.

I'll show you what happens with the first arrangement of the women.

the first arrangement is:
1234
the 2 arrangements for the men are ab and ba

stick the 2 men in the middle and you get:

12 ab 34
12 ba 34

this happens with all 24 arrangements of the women, so the total number of arrangements becomes 48.

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