Final answer:
The librarian can arrange 15 books out of 20 on a shelf in A) 465,106 different ways.
Step-by-step explanation:
To calculate the number of ways the librarian can arrange 15 books out of 20 on a shelf, we can use the concept of permutations. We need to find the number of permutations of 20 books taken 15 at a time.
The formula for permutations is given by P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items taken at a time. In this case, we have n = 20 and r = 15.
Plugging in the values, we get P(20, 15) = 20! / (20-15)! = (20 x 19 x 18 x 17 x 16) / (5 x 4 x 3 x 2 x 1) = 465,106.
Therefore, the librarian can arrange 15 books on the shelf in 465,106 different ways. So, the answer is A. 465,106.