I don't know how to do the math, but my python simulation says zero.

  import random

  service_providers = 20
  servers_threshold = 5
  up_time_threshold = 0.999**7 # prob down in week
  years = 20
  occurrences = 0

  # run sim 10,000 times
  for i in range(0, 10000):
      servers_down = False
    
      # simulate weeks in 20 years
      for j in range(1, 52*years):
          how_many_down_this_week = 0
        
          # run up times for service providers
          for s in range(0, service_providers):
              up_time = random.random()
              if up_time > up_time_threshold:
                  how_many_down_this_week += 1

          # did they go down?
          if how_many_down_this_week > servers_threshold:
              servers_down = True

      if servers_down == True:
          occurrences += 1

  print(occurrences)