KöMaL problem A, 2020

Let $k\ge t\ge 2$ positive integers. For integers $n\ge k$ let $p_n$ be the probability that if we choose $k$ from the first $n$ positive integers randomly, any $t$ of the $k$ chosen integers have greatest common divisor 1. Let $q_n$ be the probability that if we choose $k-t+1$ from the first $n$ positive integers the product is not divisible by a perfect $t$-th power that is greater then 1.