Lebanon was formed in 1920, and before that it was part of Syria in the Turkish Empire, so Lebanese people were usually listed as Syrian, or sometimes Turkish... in 1921 I would think they would be called either Syrian or Lebanese.
I see the Coreys and Nojaims in the census records living in Syracuse, NY in 1930, 1920, and the Coreys in 1910. There's some discrepancy in the census records for the year in which some of them emigrated. It says that Nicholas came over in 1900, 1903, or 1905. I would think 1903 would be most likely... in 1910 he was living with his mother, Mary S Corey, and it says that she came in 1900, and Nicholas and his brothers George and Joseph came in 1903. The others say that Ruth came in either 1911 or 1914. Her mother Nazha came in 1914, as did her brother Richard. Paul and Sadie came in 1910 or 1911, and Alexander in 1911 or 1913.
Nicholas was born about 1887, and Ruth about 1896. They got married around 1915.
If you don't have access to the census records, let me know and I can email them to you through ancestry.com.