Exercise 6

Mrs. Steffen's third grade class has $30$ students in it. The students are divided into three groups (numbered $1$, $2$, and $3$), each having $10$ students.

1. The students in group $1$ earned $10$ extra minutes of recess by winning a class competition. Before going out for their extra recess time, they form a single file line. In how many ways can they line up?

2. When all $30$ students come in from recess together, they again form a single file line. However, this time the students are arranged so that the first student is from group $1$, the second from group $2$, the third from group $3$, and from there on, the students continue to alternate by group in this order. In how many ways can they line up to come in from recess?