Use the pigeonhole principle to solve this: In a room with 565 children ages 10-12, will at least 2 children share the exact same birthday? Explain.

If a man has 4 sweaters and 5 pairs of slacks, how many different outfit can he wear?

Alice and Bill have four grandchild, and they have three framed pictures of each grandchild. They wish to choose one picture of each grandchild to display on the piano in their living room, arranged from oldest to youngest. In how many ways can they do this?

Alice and Bill have four grandchildren, and they have three framed pictures of each grandchild. They wish to choose one picture of each grandchild to display on the piano in their living room, arranged from oldest to youngest. In how many ways can they do this?