  A ribbon D over a variety C is a scheme such that D_{red} = C, the ideal I in
O_D of Cis a line bundle on C and I^2 = 0. A two dimensional ribbon is called a
carpet. In this article we show that if D is a K3 carpet, that is a ribbon
associated to a rational normal scroll with numerical invariants that of a K3
surface,then it can be smoothed to a K3 surface. We also show that not all K3
carpets are smooth points of the Hilbert scheme unlike in the dimension one
case as shown by Bayer and Eisenbud. We prove that only those K3 carpets which
are supported on "balanced" scrolls are smooth points of the Hilbert scheme.
