Mrs. Malones group of 13 students are lining up for recess. How many ways can all the students be ordered in line? How many ways can three of the students be ordered in line? Show all work

To find the number of ways all 13 students can be ordered in line, we can use the formula for permutations of a set of objects. The number of ways to arrange 'n' objects is given by n!, where n! represents n factorial.

1. Number of ways all 13 students can be ordered in line:
13! = 13 \times 12 \times 11 \times \ldots \times 2 \times 1 = 6,227,020

2. Number of ways 3 students can be ordered in line out of 13:
We can use the same formula for permutations, but this time we pick 3 students out of 13 to find the number of ways they can be arranged.
^{13}P_3 = \frac{13!}{(13-3)!} = \frac{13!}{10!} = 13 \times 12 \times 11 = 1,716

\textbf{Answer:}
1. There are 6,227,020 ways all 13 students can be ordered in line.
2. There are 1,716 ways 3 students can be ordered in line out of 13.