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

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.