Services
    .Where(s => s.Facilities.Any(f => f.Name == "Sample"))
    .GroupBy(s => s.Type)
    .Select(g => new { FacilityName = g.Key, Count = g.Count() })

Services
    .Select(s => new { Id = s.Id, Type = s.Type, Facilities = s.Facilities })
    .Where(s => s.Facilities.Any(f => f.Name == "Sample"))
    .GroupBy(s => s.Type)
    .Select(g => new { FacilityName = g.Key, Count = g.Count() })

Services
    .Select(s => new { Id = s.Id, Type = s.Type, Facilities = s.Facilities })
    //.Where(s => s.Facilities.Any(f => f.Name == "Sample"))
    .GroupBy(s => s.Type)
    .Select(g => new { FacilityName = g.Key, Count = g.Count() })