Let G be a simple graph with vertex set V\equal{}\{0,1,2,3,\cdots ,n\plus{}1\} .jand j\plus{}1 are connected by an edge for 0≤j≤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)\equal{}\{A\subset V \mid 0.n\plus{}1 \notin A,|A|\equal{}p,O(G(A))\equal{}2r\} for r≤p≤2r.
Prove That |T(p,r)|\equal{}{n\minus{}r \choose{p\minus{}r}}{n\minus{}p\plus{}1 \choose{2r\minus{}p}}. combinatorics proposedcombinatorics