WASHINGTON — First lady Michelle Obama greeted a room full of giddy high school students in the State Dining Room on Monday, as cast members from the musical “Hamilton” trailed in behind her. Performers from the wildly popular show based on the life of Alexander Hamilton were there for a day of events at the White House, and Mrs. Obama was quite giddy herself.

“We do a lot of really cool things,” she said, standing under a portrait of Abraham Lincoln in the ornate room usually used for parties. “But this, for me personally, is the coolest.”

The students laughed and nodded their heads in agreement. She called them some of the “luckiest young people on the planet,” as they got to spend the day lobbing questions at cast members from the hip-hop-inflected, historically based “Hamilton.”