Hey guys
When you have to arrange a number of items and their order is important, these arrangements are called permutations
The way to quickly calculate permutations is to use factorials
I know I just said a lot of words that dont make sense, but lets go one at a time
Factorials are a math principle where if you have an integer n, a factorial is written out as n!
n! means that you multiply all of the integers from 1 to n
If n = 5, n! = 5 * 4 * 3 * 2 * 1
The reason this is used in permutations, where you arrange items in a certain order, can be understood with simple logic
Let's say you have just three books. How many different ways can you arrange them?
In the first position, there are three different books you can place there
After you've placed the first book, you only have two books left so there are only two possible books you can place in the second position
For the third position you have no choice, you only have one book remaining to put there
How does this work out to a factorial?
Well if there are three possibilities for the first position and two for the second and one for the last, that comes to a total of 3 * 2 * 1 = 6 possible arrangements
Look:
ABC
ACB
BAC
BCA
CAB
CBA
And this principle extends for any number of items
If you had to figure out how many ways there were to arrange 700 books in order like this, you would calculate 700! Good luck!!
And the reason I specify permutations is that this is one of two ways to arrange things that are commonly tested on the GMAT, the other being what is called combinations
The first step is to understand whether you are doing a permutations problem or a combinations problem
Permutation problems are the ones where the order in which items are placed matters, like in this one, and they are way easier because you just take the factorial of the total number of items to be arranged.
In combination problems, the order the things are placed in doesn't matter and the formula is more complex
If this question was just asking how many ways there were to arrange seven books, the answer would simply be 7!
But the twist is that three of the books must stay together. That means that this three book cluster can be treated as one unit. It's the three book unit and four other books that have to be arranged. The three-book unit could go first, last or somewhere in the middle between the other four books
That means there are essentially 5 things that can be ordered -- 4 individual books, and one 3 book unit
Using our knowledge of permutations that means there are 5! ways to order these things, right?
Yes, but keep in mind that inside that 3 math book unit, there are different ways to arrange those books also
How many different ways?
3!
3 different options for position one, 2 different options for position two, and 1 option for position three
3 * 2 * 1 = 6
So there are 5! ways to arrange the 5 things and one of those five things is the three book unit. There are 3! ways to arrange the three book unit
That means there is a total of 5!3! ways to arrange all the books in this way
5! = 5 * 4 * 3 * 2 * 1 = 120
120(3!) = 120(6) = 720
The answer is (E)
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