Final answer:
There are 1,716 different collections of 7 books that can be chosen from a set of 13 books. This mathematical problem involves combinations, and it's solved using the combination formula. Additionally, the sum of probabilities of an event and its complement is always equal to 1 in probability theory.
Step-by-step explanation:
The student is asking how many different collections of 7 books can be taken out of a possible 13 books. This is a problem of combinatorics, a branch of mathematics, specifically concerning combinations. To find the number of combinations, we use the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of options (13 books), k is the number to choose (7 books), and ! denotes a factorial.
To calculate it: C(13, 7) = 13! / [7!(13-7)!] = 13! / (7!6!) = (13×12×11×10×9×8) / (6×5×4×3×2×1) = 1716. Therefore, there are 1,716 different collections of 7 books that can be chosen.
Complementary Probabilities
Regarding the sum of the probabilities of an event and its complement, it is always equal to 1. This means if an event has a certain probability of occurring, the probability that it does not occur (its complement) is 1 minus the probability of the event occurring. This is a fundamental principle in probability theory.