 # In order to keep track of circulation numbers, the library asks you to note on a form, when you leave the library, which combinations of 15 subject areas and of 8 types of material (books, current journals, databases, bound journals, videotapes, microﬁlm, microﬁche, DVDs) you used. How many possible ways are there to ﬁll in a line on the form? Ibrahim Rosales 2022-07-18 Answered
Discrete math library homework
I am working on a homework question and I am not sure if I am going about the correct way of getting to the correct answer. I feel as this is a trick question. Here is the question:
In order to keep track of circulation numbers, the library asks you to note on a form, when you leave the library, which combinations of 15 subject areas and of 8 types of material (books, current journals, databases, bound journals, videotapes, microﬁlm, microﬁche, DVDs) you used. How many possible ways are there to ﬁll in a line on the form?
I was thinking of multiplying $15\cdot 8=120$. But for some reason that did not seem correct. I was also thinking of doing ${15}^{8}$. But that number seemed too large.
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Step 1
I'll assume, however, that a typical way to fill out the form might be something like "English, History, Math; book, database, microfiche, DVD."
Step 2
That is, you specify which subjects you came for and the types of materials you used, but without going into detail as to which type of material went with which subject. In that case the number of ways to fill out the form is either $\left({2}^{15}-1\right)\left({2}^{8}-1\right)+1$ or just $\left({2}^{15}-1\right)\left({2}^{8}-1\right)$, depending one whether you do or don't include forms for people who did nothing while at the library.
###### Not exactly what you’re looking for? Javion Henry
Step 1
For each area you have either 1,2,3,...,8 number of materials:
$\left(\genfrac{}{}{0}{}{8}{1}\right)+\left(\genfrac{}{}{0}{}{8}{2}\right)+...+\left(\genfrac{}{}{0}{}{8}{8}\right)={2}^{8}-1$
You can choose 1,2,3,...,15 subjects (applying same reasoning) : ${2}^{15}-1$
Step 2
When choosing a number of subjcets you always have the same amount of ways of choosing materials so:
$\therefore \left({2}^{15}-1\right)\left({2}^{8}-1\right)$