You have many old textbooks in your apartment. You have 6 math books, 7 econ, and 4 accounting. Assume that each textbook is different than the other.

A) What is the probability that the last book is one of the economic textbooks?
B) What is the probability that none of the math books are next to eachother?

To answer the questions, we need to find the total number of possible arrangements for the books.

A) The probability that the last book is one of the economics textbooks can be found by dividing the number of favorable outcomes (arrangements where the last book is an economics textbook) by the total number of possible outcomes (all possible arrangements of the books).

Total Number of Outcomes:
Since there are 6 math books, 7 economics books, and 4 accounting books, the total number of books is 6 + 7 + 4 = 17. Therefore, there are 17 books to arrange.

Favorable Outcomes:
For the last book to be an economics textbook, we can fix one of the economics textbooks as the last book and arrange the remaining 16 books in any order. There are 7 ways to choose the last economic textbook.

Therefore, the total number of favorable outcomes is 7 × (16!).
(16! represents the arrangement of 16 books)

Substituting the values into the formula, the probability is:
P(last book is an economics book) = (7 × (16!)) ÷ (17!)

B) To calculate the probability that none of the math books are next to each other, we need to find the number of favorable outcomes where the math books are not adjacent, and then divide it by the total number of possible outcomes.

Total Number of Outcomes:
The total number of possible outcomes for this scenario is the same as the previous question, which is 17!.

Favorable Outcomes:
For the math books to not be next to each other, we can arrange the 7 economics books and 4 accounting books first with gaps between them. There would be 5 gaps available. Then, we can place the 6 math books in these gaps in a way that they are not adjacent.

The number of ways to arrange the 7 economics books and 4 accounting books is (7 + 4)!.
The number of ways to place the 6 math books in the 5 gaps is 5P6 (permutation of 6 objects in 5 available positions).

Therefore, the total number of favorable outcomes is (7 + 4)! × 5P6.

Substituting the values into the formula, the probability is:
P(none of the math books are next to each other) = ((7 + 4)! × 5P6) ÷ (17!)

A) To find the probability that the last book is one of the economic textbooks, we need to calculate the number of favorable outcomes (the last book being an econ book) and divide it by the total number of possible outcomes.

The total number of possible outcomes is the sum of all the textbooks: 6 math books + 7 econ books + 4 accounting books = 17 books.

The number of favorable outcomes is the number of econ books, which is 7.

Therefore, the probability that the last book is one of the economic textbooks is 7/17, approximately 0.4118, or 41.18%.

B) To find the probability that none of the math books are next to each other, we need to determine the number of favorable outcomes (arrangements where no math books are adjacent) and divide it by the total number of possible outcomes.

The total number of possible outcomes can be calculated by arranging all 17 textbooks, regardless of any restrictions. This can be done by finding the factorial of 17 (17!).

The number of favorable outcomes can be determined by treating the math books as a single entity and arranging the 11 entities (1 group of math books, 7 econ books, and 4 accounting books). The number of arrangements of these entities can be calculated by finding the factorial of 11 (11!).

Therefore, the probability that none of the math books are next to each other is equal to (11! / 17!).

Keep in mind that this probability may be quite small, but we can determine it mathematically.