Question

there is a pile of 8 blocks: 4 are white, 3 are yellow, and 1 is purple. By placing the blocks side by side in a straight line, how many different color patterns could you make

Answer 1

Ways to arrange 1 block: 1! = 1
Ways to arrange 3 blocks: 3! = 6
Ways to arrange 4 blocks: 4! = 24
Ways to arrange 8 blocks: 8! = 40320

The answer is 8! / (1! × 3! × 4!)

Explaination: Pretend you have the sequence WYWWWYYP. See where the 4 white blocks are? Pretend we took 2 white blocks and swapped their places. The sequence would still be WYWWWYYP. It turns out, any of the white blocks can swap places with any other white blocks and the sequence will still be the same. We know there are 4! ways to "scramble" the white blocks with only each other. So we divide by 4! In our answer. The process is the same with the yellow blocks - we divide by 3! to account for the fact that "scrambling" them produces the same sequence. And you probably saw there's only one purple block, so you can't swap it with anything and still have the same sequence - that makes sense because dividing by 1 doesn't change the answer.

Answer 2

I could make 280 different color patterns.

Explanation:

Let's suppose that we want to arrange in a straight line N objects.

Let's also suppose that in this N objects there are N1 objects of a certain class (this N1 objects are equal between them), N2 objects of a certain class (this N2 objects are also equal between them), ... , Nn objects of a certain class.

Mathematically :

`N=N1+N2+...+Nn`

This is a condition to apply the following equation

The total ways to arrange them are :

`(N!)/(N1!N2!...Nn!)`

Where ''!'' is the factorial number.

For example :

4! = 4 x 3 x 2 x 1

3! = 3 x 2 x 1

1! = 1

0! is defined as 1

In this exercise we only need to apply this equation to answer the question.

There are 8 blocks.

4 are white

3 are yellow

1 is purple

`8=4+3+1=8`

The first condition is satisfied. Now, we can apply the equation to find the arrangements.

`(8!)/(4!3!1!)=280`

There are 280 different color patterns.