A particular solution of a differential equation of order four contains how many arbitrary constants?

The general solution of an $n$th order equation contains $n$ independent arbitrary constants, so four for order four. A particular solution is obtained by assigning definite values to those constants (using initial or boundary conditions), so it contains none. Choosing $4$ confuses the particular with the general solution. The values $3$ and $1$ come from counting conditions incorrectly.