A k-pyramid \({P^k}\) is the convex hull of a \(\left( {k - {\bf{1}}} \right)\)-polytope Q and a point \({\bf{x}} \notin {\bf{aff}}\,\,Q\). Find a formula for each of the following in terms of \({f_j}\left( Q \right),j = {\bf{0}},......,n - {\bf{1}}\).

a. The number of vertices of \({P^n}\): \({f_{\bf{0}}}\left( {{P^n}} \right)\).

b. The number of k-faces of \({P^n}\): \({f_k}\left( {{P^n}} \right)\), for \({\bf{1}} \le k \le n - {\bf{2}}\)

c. The number of \(\left( {n - {\bf{1}}} \right)\) dimensional facets of \({P^n}\): \({f_{n - {\bf{1}}}}\left( {{P^n}} \right)\).

As k pyramid is the convex hull of \(\left( {k - 1} \right)\) polytope Q and \(x \notin {\rm{aff}}\,\,Q\), then the formula for vertices of \({P_n}\) is shown below:

\({f_0}\left( {{P^n}} \right) = {f_0}\left( Q \right) + 1\)

Thus, the formula is \({f_0}\left( {{P^n}} \right) = {f_0}\left( Q \right) + 1\).

The number of k-faces of \({P_n}\) is given by the formula as shown below:

\({f_k}\left( {{P^n}} \right) = {f_k}\left( Q \right) + {f_{k - 1}}\left( Q \right)\)

Thus, the formula is \({f_k}\left( {{P^n}} \right) = {f_k}\left( Q \right) + {f_{k - 1}}\left( Q \right)\).

The number of \(n - 1\) dimensional faces of \({P^n}\) is given by the formula\({f_{n - 1}}\left( {{P^n}} \right) = {f_{n - 2}}\left( Q \right) + 1\).

Thus, the formula is \({f_{n - 1}}\left( {{P^n}} \right) = {f_{n - 2}}\left( Q \right) + 1\).