Final answer:
There are 3,628,800 different ways for 10 people to line up in a cafeteria line, calculated by the factorial of 10 (10!).
Step-by-step explanation:
The question at hand is: How many ways can 10 people line up in a cafeteria line? This problem is solved by considering the number of permutations of the 10 people, as every different order is considered a unique permutation.
To find the total number of permutations, we use the formula for permutations of n distinct objects, which is n!, where "!" denotes factorial and means the product of all positive integers up to n. For 10 people lining up, it would be 10!, or 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
Calculating this, we find that there are 3,628,800 different ways for 10 people to line up in a cafeteria line.