(a) any of the 8 people can stand in any of the 8 positions from
the left-most position all the way to the right-most position.
1. Choose the person to stand left-most in 8 ways.
2. Choose the person to stand 2nd from the left as any of the
7 people who haven't yet been chosen. That can be done in 7
ways. So the first 2 positions can be filled in 8*7 ways.
3. Choose the person to stand 3rd from the left as any of the
6 people who haven't yet been chosen. That can be done in 6
ways. So the first 3 positions can be filled in 8*7*6 ways.
4. Choose the person to stand 4th from the left as any of the
5 people who haven't yet been chosen. That can be done in 5
ways. So the first 4 positions can be filled in 8*7*6*5 ways.
5. Choose the person to stand 4th from the right as any of the
4 people who haven't yet been chosen. That can be done in 4
ways. So the first 5 positions can be filled in 8*7*6*5*4 ways.
6. Choose the person to stand 3rd from the right as any of the
3 people who haven't yet been chosen. That can be done in 3
ways. So the first 6 positions can be filled in 8*7*6*5*4*3 ways.
7. Choose the person to stand 2nd from the right as either of the
2 people who haven't yet been chosen. That can be done in 2
ways. So the first 7 positions can be filled in 8*7*6*5*4*3*2 ways.
8. Choose the person to stand right-most as the 1 person who hasn't
yet been chosen. That can be done in only 1 way. So all 8
positions can be filled in 8*7*6*5*4*3*2*1 ways.
That's 8*7*6*5*4*3*2*1 = 8! = 40320 ways.
I won't go into so much detail on the others.
b) line-up if they alternate man/woman/man/….OR woman/man/woman…..?
1. Choose whether MWMWMWMW or WMWMWMWM in 2 ways.
2. Choose the men to stand in 4! ways.
3. Choose the women to stand in 4! ways.
That 2*4!*4! = 2*24*24 = 1152 ways.
c) line up if all couples stand side-by-side? _____________________________
1. Choose the left-most couple 4 ways.
2. Choose whether the man is left of the woman or
right of the woman in 2 ways.
3. Choose the next to left-most couple 3 ways.
4. Choose whether the man is left of the woman or
right of the woman in 2 ways.
5. Choose the next to right-most couple 2 ways.
6. Choose whether the man is left of the woman or
right of the woman in 2 ways.
7. Choose the right-most couple 1 way.
8. Choose whether the man is left of the woman or
right of the woman in 2 ways.
4*2*3*2*2*2*1*2 = 384 ways.
Edwin
