# Thread: How many ways can you arrange 5 things?

1. ## How many ways can you arrange 5 things?

Hi, I'm a homeschooling mom, and need help both explaining (and understanding!) my son's math problem. The problem lists five things and asks you to figure out how many different ways they could be ordered. One thing I'm sure of...there MUST be an easier way than to try and list each individual option! An answer of 120 ways is given, but I don't know how they arrived at that. Thanks in advance for any assistance you can offer!

2. ## Re: How many ways can you arrange 5 things?

Helolo, Elkay!

The problem lists five things and asks you how many different ways they could be ordered.

You are right . . . There is a better way than brute-force Listing.

Evidently you are new to "Counting Problems", so I'll baby-step through the explanation.

We assume that the five objects are distinct (different): . A, B, C, D, E.

For the first position (say, on the far left), there are 5 choices.
. . Select an object and place it there.

For the second position, there are 4 choices remaining.
. . Select an object and place it there.

For the third position, there are 3 choices remaining.
. . Select an object and place it there.

For the fourth position, there are 2 choices remaining.
. . Select an object and place it there.

For the fifth position, there is 1 object remaining.
. . Place the last object there.

$\text{Now we }multiply\text{ these numbers: }\:5\cdot4\cdot3\cdot2\cdot1 \:=\:120\text{ ways.}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

All rgua should have been explained to your son by now.

$\text{That product: }\:5\cdot4\cdot3\cdot2\cdot1\text{ can be written }\:5\,\,\text{read "five factorial".}$

So that we have, for example:

. . $\begin{array}{ccccc}4! &=& 4\cdot3\cdot2\cdot1 &=& 24 \\ 3! &=& 3\cdot2\cdot1 &=& 6 \\ 2! &=& 2\cdot1 &=& 2 \\ 1! &=& 1 &=& 1 \end{array}$

And (believe it or not): .$0!\text{ is defined to be }1.$

$\text{So for }n\text{ objects, there are: }\:n!\text{ possible orderings.}$

$\text{Your calculator may have an "n-factorial" key: }\:\boxed{n!}$

3. ## Re: How many ways can you arrange 5 things?

Thanks very much, Soroban, for your timely, thorough response. Your explanation makes sense and we've got it now! (I always did respectably well in math....I'm just a bit rusty!)

One other question, what is "rgua"? Rules Governing Universal Application????

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