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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 assertsthat 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 basedon an elementary counting method (enumerative arguments) and the application ofStirling's formula to give upper bound for some binomial coefficients.

An elemetary proof of an estimate for a number of primes less than the product of the first $n$ primes