SOLUTION: In how many ways can 6 men and 6 Women be seated together, if no two women were to sit next to each other ?

Algebra ->  Permutations -> SOLUTION: In how many ways can 6 men and 6 Women be seated together, if no two women were to sit next to each other ?      Log On


   

Question 16539: In how many ways can 6 men and 6 Women be seated together, if no two women were to sit next to each other ?
Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
In how many ways can 6 men and 6 Women be seated together, if no two women
were to sit next to each other.
`
It's easy to see that no two men can sit next to each other either. Thus men
and women must alternate seating positions
`
We can have either of these two types of arrangements
`
MWMWMWMWMWMW
`
or
`
WMWMWMWMWMWM
`
We only need to consider one of these and then multiply the answer by 2.
`
The first one
`
MWMWMWMWMWMW
`
There are 6 choices for the leftmost M
`
For each of these 6 choices for the M to seat in the 1st seat, there are 6
choices for the W to seat in the 2nd seat. That's 6×6 ways to seat the
leftmost MW.
`
For each of these 6×6 choices for the first MW in the leftmost 2 seats, there
are 5 choices for the M to seat in the 3rd seat. That's 6×6×5 ways to form
the leftmost MWM.
`
For each of these 6×6×5 choices for the first MWM in the leftmost 3 seats,
there are 5 choices for the W to seat in the 4th seat. That's 6×6×5×5 ways to
seat the leftmost MWMW.
`
For each of these 6×6×5×5 choices for the first MWMW in the leftmost 4 seats,
there are 4 choices for the M to seat in the 5th seat. That's 6×6×5×5×4 ways
to form the leftmost MWMWM.
`
For each of these 6×6×5×5×4 choices for the first MWMWM in the leftmost 5
seats, there are 4 choices for the W to seat in the 6th seat. That's
6×6×5×5×4×4 ways to form the leftmost MWMWMW.
`
For each of these 6×6×5×5×4×4 choices for the first MWMWMW in the leftmost 6
seats, there are 3 choices for the M to seat in the 7th seat. That's
6×6×5×5×4×4×3 ways to form the leftmost MWMWMWM.
`
For each of these 6×6×5×5×4×4×3 choices for the first MWMWMWM in the leftmost
7 seats, there are 3 choices for the W to seat in the 8th seat. That's
6×6×5×5×4×4×3×3 ways to form the leftmost MWMWMWMW.
`
For each of these 6×6×5×5×4×4×3×3 choices for the first MWMWMWMW in the
leftmost 8 seats, there are 2 choices for the M to seat in the 9th seat.
That's 6×6×5×5×4×4×3×3×2 ways to form the leftmost MWMWMWMWM.
`
For each of these 6×6×5×5×4×4×3×3×2 choices for the first MWMWMWMWM in the
leftmost 9 seats, there are 2 choices for the W to seat in the 10th seat.
That's 6×6×5×5×4×4×3×3×2×2 ways to form the leftmost MWMWMWMWMW.
`
For each of these 6×6×5×5×4×4×3×3×2×2 choices for the first MWMWMWMWMW in the
leftmost 10 seats, there is only 1 choice for the M to seat in the 11th seat.
That's 6×6×5×5×4×4×3×3×2×2×1 ways to form the leftmost MWMWMWMWMWM.
`
For each of these 6×6×5×5×4×4×3×3×2×2×1 choices for the first MWMWMWMWMW in
the leftmost 11 seats, there is only 1 choice for the W to seat in the 12th
seat. That's 6×6×5×5×4×4×3×3×2×2×1×1 ways to form the leftmost MWMWMWMWMWM.
`
That's 518400. But we must double this number, so the answer is 1036800.
`
Edwin
AnlytcPhil@aol.com