What is the expected number of hookups a given individual has at once?

here are n people signed up for a dating website.  For every pair of people, the website independently flips a coin that comes up head with probability p. If the coin comes up heads, the website creates a match between this pair of people. (Being a scam, the site does not actually use any specific information about the people in deciding who to hook up.) In this setting, it is perfectly possible that the website randomly creates a love triangle:

three people who are all mutually hooked up.

  1. What is the expected number of hookups a given individual has at once?
  2. What is the probability that a given individual has more than one hookup?
  3. What is the expected number of love triangles on the entire site?

 

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