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?
number of ways without restriction = C(14,5) = 2002
What we CAN'T have is no fiction or all 5 fiction
number of ways no fiction = (6,0)xC(8,5) = 56
number of ways all fiction - C(6,5)xC(8,0) = 6
no of ways of your event = 2002 - 56 - 6 = 1940
The longer way would be to find
1F, 4NF = C(6,1) x C(8,4) = 420
2F , 3NF = C(6,2) x C(8,3) = 840
3F , 2NF = C(6,3) x C(8,2) = 560
4F, 1NF = C(6,4) x C(8,1) = 120
total = 1940
To determine the number of ways the selection of books can be made, we need to consider two cases:
Case 1: When the report includes 1 fiction book
In this case, we choose 1 fiction book out of 6 and 4 nonfiction books out of 8.
Total ways = (ways to choose 1 fiction book) * (ways to choose 4 nonfiction books)
= C(6, 1) * C(8, 4)
= 6 * 70
= 420
Case 2: When the report includes 2 to 4 fiction books
In this case, we need to calculate the number of ways to choose 2, 3, or 4 fiction books and then choose the remaining nonfiction books.
Total ways = (ways to choose 2 fiction books) * (ways to choose 3 nonfiction books)
+ (ways to choose 3 fiction books) * (ways to choose 2 nonfiction books)
+ (ways to choose 4 fiction books) * (ways to choose 1 nonfiction book)
= (C(6, 2) * C(8, 3)) + (C(6, 3) * C(8, 2)) + (C(6, 4) * C(8, 1))
= (15 * 56) + (20 * 28) + (15 * 8)
= 840 + 560 + 120
= 1520
Therefore, the total number of ways to make the selection of books would be the sum of the ways from both cases:
Total ways = 420 + 1520
= 1940
So, there are 1940 ways the student can select the books for the report.
To find the number of ways the student can select the books for their report, we need to consider the restrictions given.
First, let's determine the number of ways to select 5 books from the total 14 books in the reading list.
This can be calculated using combinations, denoted as "nCr". The formula for combinations is:
nCr = n! / (r!(n-r)!)
Where n is the total number of items, and r is the number of items to be chosen.
In our case, n = 14 (the total number of books) and r = 5 (the number of books to be selected).
So, the total number of ways to select 5 books from the 14 books is:
14C5 = 14! / (5! * (14-5)!)
Simplifying this expression, we get:
14C5 = 14! / (5! * 9!)
Now, let's consider the restrictions on the types of books.
We need to ensure that at least 1 fiction book is selected and at most 4 fiction books are selected. This means that we have the following possibilities:
- 1 fiction book and 4 nonfiction books
- 2 fiction books and 3 nonfiction books
- 3 fiction books and 2 nonfiction books
- 4 fiction books and 1 nonfiction book
To find the total number of ways for each case, we need to calculate combinations for fiction books and nonfiction books separately and then multiply them together.
For each case:
1. 1 fiction book and 4 nonfiction books:
- Number of ways to choose 1 fiction book from 6: 6C1
- Number of ways to choose 4 nonfiction books from 8: 8C4
2. 2 fiction books and 3 nonfiction books:
- Number of ways to choose 2 fiction books from 6: 6C2
- Number of ways to choose 3 nonfiction books from 8: 8C3
3. 3 fiction books and 2 nonfiction books:
- Number of ways to choose 3 fiction books from 6: 6C3
- Number of ways to choose 2 nonfiction books from 8: 8C2
4. 4 fiction books and 1 nonfiction book:
- Number of ways to choose 4 fiction books from 6: 6C4
- Number of ways to choose 1 nonfiction book from 8: 8C1
Finally, we sum up the number of ways for each case to obtain the total number of ways to select the books:
Total number of ways = (6C1 * 8C4) + (6C2 * 8C3) + (6C3 * 8C2) + (6C4 * 8C1)
We can now calculate this expression to find the answer.