Word Problem

[rpdrake]

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?

 

[galactus]

Let's see. Try this.

\(\displaystyle \L\\\sum_{k=1}^{4}{{6}\choose{k}}\cdot{{8}\choose{5-k}}\)

 

[soroban]

Hello, rpdrake!

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.

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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.