MathDB

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 qn 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

.
Prove that sequences

p_n

and

q_n

converge to the same value.

A. 785