Going to the library

Probability Level 1

You go to check out three books at the library, and you want one history book, one science book, and one fantasy book.

The library has 50 history books, 95 fantasy novels, and 30 books about science. How many combinations of books do you have to choose from?

Unknown 175 145000 15000 142500

6 solutions

Karleigh Moore
Apr 18, 2016

We can solve this using the rule of product . You know that you have three categories you need to choose from. There are 50 options for the first category (history books), 95 options for the second category (fantasy books), and 30 options for the third category (science books). The rule of product states that if there are $n$ ways of doing one thing and $m$ ways of doing another thing, then there are $n \times m$ ways of doing both things. We can extend this for three options, $50 \times 95 \times 30 = 142500$ .

rule of product is still not clear to me :(

Nsdjtsgo Onm - 4 years, 10 months ago

Recall in the rule of sum its only applies to choices that are mutually exclusive, meaning to say, they cannot both occur at the same time. For example, the set of outcomes of the single coin toss, which can result in either heads or tails, but not both.

But in the rule of product the two or more actions can be done at the same time. For example, you want to wear your shirts and you have 10 different colors. And your pants and you have 6 different colors. You can choose 1 shirt and 1 pants, rule of product can be applies, and you have 10*6=60 different outfits.

Endo Genic - 2 years, 11 months ago

I find it helpful to imagine rule of product geometrically (or as a matrix): with 2 categories of options (say, 50 history books and 95 fantasy books) you get a rectangle with such lengths (50×95) and it is filled with resulted options at intersections — each row is an option of a particular fantasy book with every history book and each column is an option of a particular history book with every fantasy book. Same with 3 categories, but now it’s a rectangular cuboid instead of rectangle.

Arthur T - 1 year ago

Damn you must be an engineer or something! Good for you thinking in this way. I just use the branches they explained with, that made it super easy for me

Desanka Dimitrova - 11 months, 2 weeks ago

The rule of product can be applied to this sum like this

for 1 of the 95 fantasy novels there will be 50 history books

total no. of cases = 95* (1 * 50) = 95*50

for 1 fantasy novel and 1 history book there will 30 books on science

total no. of cases = 95 * 50 * (1 * 30) = 95 * 50 * 30

this is how I understood as I am a beginner

hope this helps

Dishita Meshtru - 10 months, 2 weeks ago

I get it now!!

Darryl Shepherd - 11 months, 2 weeks ago

I get it now!! You just keep multiplying the options

Darryl Shepherd - 11 months, 2 weeks ago

Thank you soo much 😁!

Alexandra Schmidt - 11 months, 1 week ago

woah thanks

Joshua Amara - 9 months, 3 weeks ago

I love thinking about this as a rectangle, geometric representations are the best IMO, thank you for the assist!

Amber Dantzler - 7 months, 1 week ago

There is a restriction : m is not the same for N possibilities ( m1= 30 science books; m2=95 fantasy books ; m3=50 history books )

aimad sale - 4 months, 3 weeks ago

Kauman Kauma
Aug 24, 2018

We can use the answers to quickly determine the correct choice. Using the rules of product, we know that there must be 2 zeroes at the end of the answer when we multiply 50 × 95 × 30.

Given that one of the numbers is correct, 142500 must be the right answer.

By your logic, you would immediately know that there is only 1 possible answer... But even then, it is not enough to look at how many zeros there are in the n, m, x values. What if some of them multiplies 2 x 5 somewhere in the long multiplication? That could add another zero. You cannot cheat in logic I guess

Desanka Dimitrova - 11 months, 2 weeks ago

Danish Asghar
Apr 10, 2017

50 history books 95 fantasy novels 30 books about science

rules of product = 50 * 95 * 30 = 142500

Luis Gerardo Sanchez Robledo
Jun 3, 2021

I got it this way, say I pick 1 of the 95 fantasy books, then I pick 1 of the 50 history books, once I'm done I pick 1 of the 30 science books, that would be one option. So I could do this 95 times and every time I will have the same 50 options for the history books and then another 30 for the science books, therefore I say that 95 x 50 = 4750 is the number of options for each one of the fantasy and history books together, and then 4750 x 30 = 142500 is the total amount of options for the three types of books together. Hope it helps.

Halsey Chou
May 4, 2020

Since there are 50 options for choosing history book,95 options for choosing fantasy book and 30 options for choosing science book. In order to solve the problem we must get acquainted with ourselves to the rule of product which states that if there are n ways of doing one thing and there's m ways of doing another thing ,then there are n m ways of doing both things .We can solve this by extending it to our three options,50 30*95=142500

Aaron Ma
Jan 19, 2020

We can solve this through the counting principle. 50 history books * 95 fantasy novels * 30 books on science = 142500 combinations of books to choose from.

Going to the library

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