The reading list for a literature class has 7 fiction and 9 nonfiction books. A student is to write a report on 5 books from the list during the semester. The reports must include at least 1 and at most 4 reports of fiction books. In how many ways can the selection of books be made?
Give it a try.
I showed you how to do two of these now, follow my method
576
To find the number of ways to select the books, we need to consider the different combinations that satisfy the given conditions.
We can break down the problem into cases based on the number of fiction books chosen for the report.
Case 1: Selecting 1 fiction book:
There are 7 fiction books to choose from and we need to choose 1. The number of ways to do this is C(7, 1) = 7.
Case 2: Selecting 2 fiction books:
There are 7 fiction books to choose from and we need to choose 2. The number of ways to do this is C(7, 2) = 21.
Case 3: Selecting 3 fiction books:
There are 7 fiction books to choose from and we need to choose 3. The number of ways to do this is C(7, 3) = 35.
Case 4: Selecting 4 fiction books:
There are 7 fiction books to choose from and we need to choose 4. The number of ways to do this is C(7, 4) = 35.
Now, let's consider the number of ways to select the nonfiction books. Since we have to write a report on 5 books in total and at most 4 reports of fiction books, we need to select the remaining books from the nonfiction list.
Case 1: Selecting 4 nonfiction books:
There are 9 nonfiction books to choose from and we need to choose 4. The number of ways to do this is C(9, 4) = 126.
Case 2: Selecting 3 nonfiction books:
There are 9 nonfiction books to choose from and we need to choose 3. The number of ways to do this is C(9, 3) = 84.
Case 3: Selecting 2 nonfiction books:
There are 9 nonfiction books to choose from and we need to choose 2. The number of ways to do this is C(9, 2) = 36.
Case 4: Selecting 1 nonfiction book:
There are 9 nonfiction books to choose from and we need to choose 1. The number of ways to do this is C(9, 1) = 9.
Finally, we need to multiply the number of ways for each case to find the total number of ways to select the books:
Case 1: 7 * 126 = 882
Case 2: 21 * 84 = 1764
Case 3: 35 * 36 = 1260
Case 4: 35 * 9 = 315
Total number of ways to select the books = 882 + 1764 + 1260 + 315 = 4221.
Therefore, there are 4221 ways to select the books for the report.