$\mathrm{A}$set $B$contain 2007 elements. Let$C$ be the set con sisting  of subsets of$B$ which contain at most $1003$

elements. The number of elements in $C$ is

 The number of required subsets is 

$\begin{array}{l}\mathrm{N}{=}^{2007}{C}_0{+}^{2007}{C}_1{+}^{2007}{C}_2+\dots {+}^{2007}{C}_{1003}\\ \mathrm{N}{=}^{2007}{C}_{2007}{+}^{2007}{C}_{2006}+\dots {+}^{2007}{C}_{1004}\end{array}$

Adding above two equations we get  

$\begin{array}{l}2\mathrm{N}{=}^{2007}{C}_0{+}^{2007}{C}_1+\dots {+}^{2007}{C}_{2007}\\ \Rightarrow \mathrm{N}=2^{2006}\end{array}$