Given a set of elements N = {1, 2, ....., n} and two arbitrary subsets $$A\, \subseteq \,N\,$$ and $$B\, \subseteq \,N\,$$, how many of the n! permutations $$\pi $$ from N to N satisfy $$\min \,\left( {\pi \,\left( A \right)} \right) = \min \,\left( {\pi \,\left( B \right)} \right)$$, where min (S) is the smallest integer in the set of integers S, and $${\pi \,\left( S \right)}$$ is the set of integers obtained by applying permutation $${\pi}$$ to each element of S?

What are the eigen values of the matrix $$P$$ given below? $$$P = \left( {\matrix{ a & 1 & 0 \cr 1 & a & 1 \cr 0 & 1 & a \cr } } \right)$$$

$$F$$ is an $$n$$ $$x$$ $$n$$ real matrix. $$b$$ is an $$n$$ $$x$$ $$1$$ real vector. Suppose there are two $$n$$ $$x$$ $$1$$ vectors, $$u$$ and $$v$$ such that $$u \ne v$$, and $$Fu = b,\,\,\,\,Fv = b$$

Which one of the following statements is false?

Consider the undirected graph $$G$$ defined as follows. The vertices of $$G$$ are bit strings of length $$n$$. We have an edge between vertex $$u$$ and vertex $$v$$ if and only if $$u$$ and $$v$$ differ in exactly one bit position (in other words, $$v$$ can be obtained from $$u$$ by flipping a single bit). The ratio of the choromatic number of $$G$$ to the diameter of $$G$$ is