Krdytkyu

0 $begingroup$ Let $K$ be a field. Prove that every element in $K(x)backslash K$ is transcendental over $K$ . Is proof of the question above similar to that of the question below? field-theory transcendental-numbers share | cite | improve this question edited Dec 10 '18 at 23:36 Leyla Alkan asked Dec 10 '18 at 23:29 Leyla Alkan Leyla Alkan 1,575 1 7 24 $endgroup$

6 3 $begingroup$ This question is inspired by the following problem: Randomly place seven balls into ten bins, with no bin containing more than one ball. What is the probability that there will be (at least) two consecutive bins that are empty? Note that this is not a circular arrangement, although that could make for an interesting (if simpler) question. The actual question I have is the same as the one above, but with the numbers written in emphasized text generalized as positive integers $n$ , $k$ , and $m$ , respectively. Question: Randomly place $n$ balls into $k$ bins, with no bin containing more than one ball. What is the probability that there will be (at least) $m$ consecutive bins that are empty? I suspect that this is a known counting problem (as framed, or through some equivalen