How many minimum spanning tress does the following graph have? Draw them (Weights are assigned to the edges).

Prove that in a finite graph, the number of vertices of odd degree is always even.

The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is $$$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr . & . & . & . & . & . & . \cr . & . & . & . & . & . & . \cr . & . & . & . & . & . & . \cr 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr } } \right]$$$

The probability that a number selected at random between $$100$$ and $$999$$ (both inclusive ) will not contain the digit $$7$$ is