An English teacher needs to pick 13 books to put on her reading list for the next school year, and she needs to plan the order in which they should be read. She has narrowed down her choices to 5 novels, 7 plays, 7 poetry books, and 5 nonfiction books.
Step 1 of 2 : If she wants to include no more than 2 nonfiction books, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.
5C2 * 21P13
1st answer includes 2 nonfiction
she could also include one or none
one ... 5 * 20P13
none ... 19P13
To calculate the number of different reading schedules, we need to consider the number of ways the nonfiction books can be selected.
Step 1: Choose 0 nonfiction books:
In this case, she can choose 13 books from the 19 remaining books (5 novels, 7 plays, 7 poetry books) in 19C13 ways.
Step 2: Choose 1 nonfiction book:
In this case, she can choose 12 books from the 14 remaining books (5 novels, 7 plays, 7 poetry books, and 1 nonfiction book) in 14C12 ways.
Step 3: Choose 2 nonfiction books:
In this case, she can choose 11 books from the 9 remaining books (5 novels, 7 plays, 7 poetry books, and 2 nonfiction books) in 9C11 ways.
Therefore, the total number of different reading schedules is:
19C13 + 14C12 + 9C11 = 86465 + 91 + 9 = 86565
Expressing this in scientific notation, rounding to the hundredths place gives 8.66 x 10^4.
So, there are 8.66 x 10^4 different reading schedules possible.
To find the number of different reading schedules possible, we need to consider the number of ways the books can be arranged.
First, we need to select the 13 books from the given choices. Since the teacher wants to include no more than 2 nonfiction books, there are two cases to consider:
1. Selecting 0 nonfiction books: In this case, we need to select 13 books from the remaining choices of 5 novels, 7 plays, and 7 poetry books.
2. Selecting 1 nonfiction book: In this case, we need to select 12 books from the remaining choices, which are 4 novels, 7 plays, and 7 poetry books.
Let's calculate the number of ways for each case:
1. Selecting 0 nonfiction books:
We need to choose 13 books out of (5 novels + 7 plays + 7 poetry books). This can be calculated using the combination formula as C(19, 13).
C(19, 13) = 19! / (13! * (19-13)!) = 19! / (13! * 6!)
2. Selecting 1 nonfiction book:
We need to choose 1 nonfiction book and 12 books out of (4 novels + 7 plays + 7 poetry books). This can be calculated as C(5, 1) * C(18, 12).
C(5, 1) * C(18, 12) = (5! / (1! * (5-1)!)) * (18! / (12! * (18-12)!)) = (5 * 4 * 3 * 2 * 1) * (18! / (12! * 6!))
Now, let's calculate both cases and find the total number of possible reading schedules:
Total = C(19, 13) + C(5, 1) * C(18, 12)
Calculating this value will give us the answer.