Combination and Probability Together: Meredith has twelve school books to fit in....

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TreasureDragon

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I've tried getting help elsewhere and find the solution for this but I just can't find an example similar to this one.

Problem: Meredith has twelve school books to fit in her shelves. Four of these books are math books and six of the books are language arts. In how many distinct ways can Meredith arrange the books in a single row on her shelf if she keeps all the math books together and all of the language arts books together?
 
TreasureDragon said:
Problem: Meredith has twelve?(ten? 4+6) school books to fit in her shelves. Four of these books are math books and six of the books are language arts. In how many distinct ways can Meredith arrange the books in a single row on her shelf if she keeps all the math books together and all of the language arts books together?
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Let's play the back-of-the-book game. It gives the answer as \(\displaystyle 2\cdot 4!\cdot 6!\).
Now you must write back and explain why that is correct.

BTW(that answer is for ten books not twelve)
 
First, we have to find out in how many ways she can keep all the math books together and all of the language arts books together. Try to solve this, and after we will discuss the second part of the problem.
 
Start by thinking of this as putting four items on the shelf- thinking of the four math books as one item, the six language books as a second item, and the two other books as the third and fourth. You have 4 choices of which of those to put on the shelf first, then three choices for the second, the two choices for the third, and the one left last. There are 4(3)(2)(1)= 4!= 24 ways to do that.

Now, how many ways are there to rearrange just the four math books? The six language books?
 
So, since we have 24 ways to combine them assuming that Math was one big book and LA being one big book, I guess we can multiply this by Math: 4! and LA: 6! for a total of 414,720 combinations, which is the right answer! Thank you all! :)
 
TreasureDragon said:
So, since we have 24 ways to combine them assuming that Math was one big book and LA being one big book, I guess we can multiply this by Math: 4! and LA: 6! for a total of 414,720 combinations, which is the right answer!
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No that is not correct.

\(\displaystyle 2!\cdot 4!\cdot 6!=34560\) is correct. See here.
 
pka said:
No that is not correct.

\(\displaystyle 2!\cdot 4!\cdot 6!=34560\) is correct. See here.
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Prof, the student (not very clearly) stated that there were 12 books--4 math books, 6 language arts books and two others books of different types. So yes the number of ways to arrange the books given that .... is 4!(4!*6!)= 414720
 
pka said:
No that is not correct.

\(\displaystyle 2!\cdot 4!\cdot 6!=34560\) is correct. See here.
Click to expand...

Apologies for the confusion there. I did not want to misinterpret anything over here so I have copy-pasted the exact question as to avoid any user-bias within the question (the way I understood it).
 
Jomo said:
Prof, the student (not very clearly) stated that there were 12 books--4 math books, 6 language arts books and two others books of different types. So yes the number of ways to arrange the books given that .... is 4!(4!*6!)= 414720
Click to expand...
Here is the OP.
TreasureDragon said:
Problem: Meredith has twelve school books to fit in her shelves. Four of these books are math books and six of the books are language arts. In how many distinct ways can Meredith arrange the books in a single row on her shelf if she keeps all the math books together and all of the language arts books together?
Click to expand...
But if you read that into the statement then there are many other readings.
If she has eight art books then she has six, does she not?
Or if she has six mathematics books then she has four, does she not?

If a person has seven siblings then it is true that the person has four siblings.
That is why test company pay big bucks to have their test edited. If I were reading this question, I would have insisted that it be changed to, "ten books or including two books of other subjects"

Reading the OP, how are we to decide if the twelve is a typo, or else it is just a sloppy question.
In general the rule-of-thumb is to assume a typo.
 
pka said:
Here is the OP.

But if you read that into the statement then there are many other readings.
If she has eight art books then she has six, does she not?
Or if she has six mathematics books then she has four, does she not?

If a person has seven siblings then it is true that the person has four siblings.
That is why test company pay big bucks to have their test edited. If I were reading this question, I would have insisted that it be changed to, "ten books or including two books of other subjects"

Reading the OP, how are we to decide if the twelve is a typo, or else it is just a sloppy question.
In general the rule-of-thumb is to assume a typo.
Click to expand...
I based my response on the answer that the OP claimed was right along with a very very sloppy original post. I just don't want the student to be confused. You of course are correct that the problem should have either stated 10 books or with two books of different subjects.
 
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