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### Explaining Permutation to Young Students

```Date: 01/20/2007 at 11:42:41
From: Daniel
Subject: How to explain permutation to a kid

Question:  How many ways can we arrange 4 people A, B, C, D to sit in
a different order?

Answer:  For the first seat, we have 4 choices, for the second seat,
we have 3 choices, for the third seat, we have 2 choices, and for the
last seat, 1 choice.  Therefore, the answer is 4 x 3 x 2 x 1 = 24.

My kid wants to know why it's not 4 + 3 + 2 + 1 = 10.  I'm having a
hard time explaining this to him.

```

```
Date: 01/20/2007 at 20:31:28
From: Doctor Peterson
Subject: Re: How to explain permutation to a kid

Hi, Daniel.

I like to add a few extra words in the explanation.  As you stated the
answer (which is the way it's commonly done), it does sound as if you

Here's how I say it:

You have 4 ways to choose a person for the first seat. Then, FOR
EACH of those 4 choices, there are 3 ways to choose a person for
the second seat. Then FOR EACH of the 12 choices you've made so
far, you have 2 ways to choose the person to sit in the third seat,
making a total of 24 ways to choose. The last seat leaves us no
choice, since there is only 1 person left to pick.

If we try actually making a list of the possible orders, we can watch
this happening. After filling the first seat, we have these 4 possible
orders:

A _ _ _

B _ _ _

C _ _ _

D _ _ _

Then for each case, we have three ways to fill in the second seat:

A B _ _    A C _ _    A D _ _

B A _ _    B C _ _    B D _ _

C A _ _    C B _ _    C D _ _

D A _ _    D B _ _    D C _ _

Then, for each of those 24 choices, we still have two ways to fill in
the third seat:

A B C _    A C B _    A D B _
A B D _    A C D _    A D C _

B A C _    B C A _    B D A _
B A D _    B C D _    B D C _

C A B _    C B A _    C D A _
C A D _    C B D _    C D B _

D A B _    D B A _    D C A _
D A C _    D B C _    D C B _

The last seat fills itself:

A B C D    A C B D    A D B C
A B D C    A C D B    A D C B

B A C D    B C A D    B D A C
B A D C    B C D A    B D C A

C A B D    C B A D    C D A B
C A D B    C B D A    C D B A

D A B C    D B A C    D C A B
D A C B    D B C A    D C B A

So there are the 24 ways to seat the four people, which we found by
making 4 times 3 times 2 times 1 choices.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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