The reading list for a literature class has six fiction and eight nonfiction books. A student is to write a report on five 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?
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
Word Problem
- Thread starter rpdrake
- Start date
Hello, rpdrake!
I have a back-door approach to this problem.
There are: \(\displaystyle \:{14\choose5} \:=\:2002\) possible selections.
The restriction (one to four fiction books) means both types must be included.
The only cases which are not included are: "all fiction" and "all nonfiction".
. . There are: \(\displaystyle \:{6\choose5} \:=\:6\) ways to choose all fiction.
. . There are: \(\displaystyle \:{8\choose5} \:=\:56\) ways to choose all nonfiction.
. . Hence, there are: \(\displaystyle \,6\,+\,56\:=\:62\) selections which are not[/i] allowed.
Therefore, there are: \(\displaystyle \:2002\,-\,62 \:=\:\fbox{1940}\) permitted selections.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
To double-check, I also solved the problem head-on.
There are four permitted selections . . .
. . 1 fiction, 4 nonfiction . There are: \(\displaystyle \:{6\choose1}{8\choose4} \:=\:420\) ways.
. . 2 fiction, 3 nonfiction . There are: \(\displaystyle \:{6\choose2}{8\choose3} \:=\:840\) ways.
. . 3 fiction, 2 nonfiction . There are: \(\displaystyle \:{6\choose3}{8\choose2} \:=\:560\) ways.
. . 4 fiction, 1 nonfiction . There are: \(\displaystyle \:{6\choose4}{8\choose1} \:=\:120\) ways.
Therefore, there are: \(\displaystyle \:420\,+\,840\,+\,560\,+\,120\;=\;\fbox{1940}\) permitted selections.
I have a back-door approach to this problem.
The reading list for a literature class has 6 fiction and 8 nonfiction books.
A student is to write a report on five 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?
There are: \(\displaystyle \:{14\choose5} \:=\:2002\) possible selections.
The restriction (one to four fiction books) means both types must be included.
The only cases which are not included are: "all fiction" and "all nonfiction".
. . There are: \(\displaystyle \:{6\choose5} \:=\:6\) ways to choose all fiction.
. . There are: \(\displaystyle \:{8\choose5} \:=\:56\) ways to choose all nonfiction.
. . Hence, there are: \(\displaystyle \,6\,+\,56\:=\:62\) selections which are not[/i] allowed.
Therefore, there are: \(\displaystyle \:2002\,-\,62 \:=\:\fbox{1940}\) permitted selections.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
To double-check, I also solved the problem head-on.
There are four permitted selections . . .
. . 1 fiction, 4 nonfiction . There are: \(\displaystyle \:{6\choose1}{8\choose4} \:=\:420\) ways.
. . 2 fiction, 3 nonfiction . There are: \(\displaystyle \:{6\choose2}{8\choose3} \:=\:840\) ways.
. . 3 fiction, 2 nonfiction . There are: \(\displaystyle \:{6\choose3}{8\choose2} \:=\:560\) ways.
. . 4 fiction, 1 nonfiction . There are: \(\displaystyle \:{6\choose4}{8\choose1} \:=\:120\) ways.
Therefore, there are: \(\displaystyle \:420\,+\,840\,+\,560\,+\,120\;=\;\fbox{1940}\) permitted selections.