Problem with people sitting in the train

Question

I am supposed to solve the problem:

Ten married couples travel by train in four wagons. How many ways can these 20 people travel if at least one of them has to sit in each wagon, and no married couple wants to travel in the same wagon?

My solution:

$20.18.16.14$ because we take every other person, but that is not correct answer. Or should I use inclusion-exclusion? Can anyone tell me, where is the problem in my solution?

Answer 1

1. Number of ways to sit 20 married couples in 0 wagons, no married couple in the same wagon, possibly leaving some wagons empty: 0
2. Number of ways to sit 20 married couples in 1 wagons, no married couple in the same wagon, possibly leaving some wagons empty: 0
3. Number of ways to sit 20 married couples in 2 wagons, no married couple in the same wagon, possibly leaving some wagons empty: $2^{10}$
4. Number of ways to sit 20 married couples in 3 wagons, no married couple in the same wagon, possibly leaving some wagons empty: $6^{10}$
5. Number of ways to sit 20 married couples in 4 wagons, no married couple in the same wagon, possibly leaving some wagons empty: $12^{10}$
6. Number of ways to sit 20 married couples in 4 wagons, no married couple in the same wagon, no wagon empty:

$$ 12^{10} - 4 \times 6^{10} - 6 \times 2^{10} + 4 \times 0 - 1 \times 0 = 61675505664.$$