The reading list for a literature class has 6 fiction and 8 nonfiction books. A student is to write a report on 5 books from the list during the semester. The report must include at least 1 and at most 4 reports of fiction books. In how many ways can the selection of books be made?
To find the number of ways the student can select books, we can break down the problem into different cases based on the number of fiction books chosen for the report.
Case 1: Selecting 1 fiction book and 4 nonfiction books:
The student can choose 1 fiction book from the available 6 in C(6, 1) ways, and 4 nonfiction books from the available 8 in C(8, 4) ways. Therefore, the number of ways for this case is C(6, 1) * C(8, 4).
Case 2: Selecting 2 fiction books and 3 nonfiction books:
The student can choose 2 fiction books from the available 6 in C(6, 2) ways, and 3 nonfiction books from the available 8 in C(8, 3) ways. Therefore, the number of ways for this case is C(6, 2) * C(8, 3).
Case 3: Selecting 3 fiction books and 2 nonfiction books:
The student can choose 3 fiction books from the available 6 in C(6, 3) ways, and 2 nonfiction books from the available 8 in C(8, 2) ways. Therefore, the number of ways for this case is C(6, 3) * C(8, 2).
Case 4: Selecting 4 fiction books and 1 nonfiction book:
The student can choose 4 fiction books from the available 6 in C(6, 4) ways, and 1 nonfiction book from the available 8 in C(8, 1) ways. Therefore, the number of ways for this case is C(6, 4) * C(8, 1).
Now, we can sum up the number of ways for all the cases to get the total number of ways:
Total number of ways = C(6, 1) * C(8, 4) + C(6, 2) * C(8, 3) + C(6, 3) * C(8, 2) + C(6, 4) * C(8, 1)
Calculating these combinations, we have:
Total number of ways = 6 * 70 + 15 * 56 + 20 * 28 + 15 * 8
Total number of ways = 420 + 840 + 560 + 120
Total number of ways = 1940
Therefore, there are 1940 different ways for the student to select the books for the report.