Given a number n. Suppose $p\_1< p\_2<....< p\_k$ be any k distinct primes less than n. How do i find the cardinality of the set, S={$(e\_1,e\_2,...,e\_k): p\_1^{e\_1}p\_2^{e\_2}...p\_k^{e\_k} < n$}. Assume that $(1,1,...,1)\in S$ and $e\_i\geq 1\;\;\forall i$. It will be great if one could give a non-trivial upper bound for the cardinality of this set.

P.S. I have a way of getting the cardinality of the set approximately for four primes but things get complicated when the number of primes are increased.