  We consider the k-th order statistic from unit exponential distribution and
show that it can be represented as a sum of independent exponential random
variables. Our proof is simple and different. It readily proves that the
standardized exponential spacings also follow unit exponential distribution. An
interesting probabilistic proof of the Basel problem is also given. Another
advantage of our approach is that by computing the Laplace transform of the
k-th order statistic in two different ways, we derive several interesting
combinatorial identities. A probabilistic interpretation of these identities
and their generalizations are also given.
