If 5 parts of a shoe is customisable, and you can choose from 12 colors for each part. How many different shoe is possible ?

Question

Hi , I can't figure this problem out. if somebody can solve this , that would be amazing. a short description on how it's done is appreciated but optional .

Answer 1

There are 5 independent parts of a shoe.

Each part may have one of 12 unique colors.

 

First, think of the number of combinations on a counter (like a gas pump, an odometer, a combination lock, etc.) that allows digits 0-9 (that is 10 unique digits):

      D1      D2     D3

       0        0        0

       1        1        1

       2        2        2

       3        3        3

       4        4        4

       5        5        5

       6        6        6

       7        7        7

       8        8        8

       9        9        9

 

This counts from 000 to 999.  That is 1000=10*10*10 numbers.

 

Now, the shoe has 5 parts.  Each part may have one of 12 unique colors.  We may count them the same way:

      P1      P2      P3      P4       P5

       1        1        1        1        1

       2        2        2        2        2

       3        3        3        3        3

       4        4        4        4        4

       5        5        5        5        5

       6        6        6        6        6

       7        7        7        7        7

       8        8        8        8        8

       9        9        9        9        9

      10      10      10      10      10

      11      11      11      11      11

      12      12      12      12      12

 

But now we have 12*12*12*12*12 = 12⁵ = 248832 possible values.  PLZ note that there will be 12 shoes with all the same color (since there are 12 colors) and lots of shoes with multiple parts of the same color. 

Answer 2

We first need to find out how many arrangements can be done with the shoe and the color. A shoe with 5 parts can be customizable 120 ways. This is done by using the factorial of 5. [5x4x3x2x1] = 5!

The same thing with the colors. 12! = 479001600 ways.

 

Think of it like this. The colors are limited to how many parts of a shoe. 

So color red can appear 120 ways on the shoe. How? You may ask. Red does not just have to appear once; it can appear twice or even 5 times. Eg: Part1=Red,Part2=Red,Part3=Red,Part4=Red,Part5=Red.

Part1=Red,Part2=Red,Part3=Orange,Part4=Orange,Part5=Blue.

 

Also, each part of the shoe is independent from the other. You have to treat them as their own combination.

 

But if we want to find out how many variations of both there are, we need to use the permutation formula which is P(n,r) = n!/[(n-r)!]. n is the number of different items and r is the number of them appearing in an arrangement. This makes it SO Much easier than trying to count every single combination. Therefore, it is P(12,5). Just plug in into your calculator. Hope this helps.