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There are many songs that are considered "Christmas carols" and tell the story of the birth of Jesus Christ. Here are three examples:
"O Come, All Ye Faithful" is a traditional Christmas carol that tells the story of the birth of Jesus. It describes the scene of the shepherds visiting the stable where Jesus was born, and praises the newborn King for coming to save humanity.
"Hark! The Herald Angels Sing" is another traditional Christmas carol that tells the story of the birth of Jesus. It describes the angel Gabriel appearing to Mary to announce that she will be the mother of Jesus, and the shepherds hearing the good news of Jesus' birth from the angels.
"O Little Town of Bethlehem" is a Christmas carol that tells the story of Jesus' birth from the perspective of the town of Bethlehem. It describes the town as it was on the night that Jesus was born, with the people "wrapped in sleep" and the stars shining brightly in the sky. It also mentions the arrival of the wise men who came to worship the newborn King.
These three Christmas carols are just a few examples of the many songs that tell the story of the birth of Jesus. There are many other Christmas carols that tell this story in different ways and with different details.