  Let $\alpha$ be a real number such that $1< \alpha <2$ and let
$x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$
x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$
  Then we prove that for each positive integer $n>x_0$ there exist at least
$[n^\alpha]$ primes between the $(n+1)$th prime and the product of the first
$n+1$ primes. In particular, we establish a recent Cooke's result which asserts
that for each positive integer $n$ there are at least $n$ primes between the
$(n+1)$th prime and the product of the first $n+1$ primes. Our proof is based
on an elementary counting method (enumerative arguments) and the application of
Stirling's formula to give upper bound for some binomial coefficients.
