Final answer:
The number of different ways to choose and rank the three best movies out of ten is calculated using permutations, with the formula 10P3 = 10! / (10-3)! The result is 720 different ways to rank the movies.
Step-by-step explanation:
The student is asking about the number of different ways to choose and rank the three best movies out of ten seen during the year. This is a problem of permutations because the order in which the movies are ranked matters.
To solve this, we use the formula for permutations of choosing and arranging r objects from n total objects, which is nPr = n! / (n-r)!, where ! denotes factorial. In this case, we want to arrange 3 movies (r=3) out of a total of 10 (n=10).
Calculating this gives:
10P3 = 10! / (10-3)!
= 10! / 7!
= (10 × 9 × 8 × 7!) / 7!
= 10 × 9 × 8
= 720.
Therefore, there are 720 different ways to choose and rank the three best movies from ten.