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Random graphs- Erdos and Renyi 1959 paper

Please refer to this link. It is Erdos and Renyi's first paper on Random Graphs (1959). I am trying to work through it.

I'm struggling with equations (16), (17) and (21).

(16)

  • I'm not sure why they are using those two bounds in the summation? It seems redundant
  • Why they did not use the result from (15) in (16) in the summation. They only used (14)?

(17)

  • Why does loglog(n) replace M in (17)?

(21)

  • What does the (-1)^k figure do? Why is it necessary?