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

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Let α be a real number such that 1< α<2 and let x₀=x₀(α) be a (unique) positive solution of the equation xα⁻¹ -πe²√3x +1=0. Then we prove that for each positive integer n>x₀ there exist at least [nα] 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.