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
would just add the numbers.

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/