IN-TST 2009 Problem 12

Let $G$ be a simple graph with vertex set $V=\{0,1,2,3,\cdots ,n+1\}$ . $j$ and $j+1$ are connected by an edge for $0\le j\le n$ . Let $A$ be a subset of $V$ and $G(A)$ be the induced subgraph associated with $A$ . Let $O(G(A))$ be number of components of $G(A)$ having an odd number of vertices. Let $T(p,r)=\{A\subset V \mid 0.n+1 \notin A,|A|=p,O(G(A))=2r\}$ for $r\le p \le 2r$ . Prove That $|T(p,r)|={n-r \choose{p-r}}{n-p+1 \choose{2r-p}}$ .