Combinatory Problem, all the different ways to do this combinatory problem

Question

if there are 10 differentiable boxes and you need to paint all them and dry the paint. how many different ways there are to do it? You need to paint the box to dry the paint. For example if you paint the first box you can`t dry the second box you need to paint the second box to dry the second box. I am having a lot of problems with this problem, i thought the solution was 10!+10!. i know that:

• With 1 box the solution is 1
• With 2 boxes the solution is 6
• With 3 boxes the solution is 90

Answer 1

The solution is 20! / (2^10), because the average needs to be split

Answer 2

Hint: Compare this to the problem of counting how many sequences of length $2n$ use each of the number $1,2,\dots,n$ exactly twice each.

The first occurrence of a number corresponds to that box being painted at that time. The second occurrence of the same number corresponds tot hat box being dried at that time.

For example $\underline{1}~\underline{1}~\underline{2}~\underline{2}$ corresponds to the first box being painted, then dried, then the second box being painted then dried while the sequence $\underline{2}~\underline{1}~\underline{2}~\underline{1}$ corresponds to the second box being painted, then the first box being painted, then the second box being dried then the first box being dried.