# How to calculate the possible number of combinations

1. Mar 10, 2015

### Laura72

Hi, I am new here hoping to find someone who can help.

I am an artist working mainly in Geometric forms. My work follows a set grid and I want to use math to work out all the possible combinations of my grid. Math generally makes my head hurt, so I am hoping someone can explain in real simple terms how to work this out!

My grid is a square split into 16 sections. I would like to know how many possible combinations could be made by coloring one section, or combinations of more than one section.

E.g. I know I could produce 16 paintings by colouring one different section on each painting and leaving the others blank. But what if I colour sections 1 and 8 and leave the others blank, or sections 1, 8 and 11….etc.

How many possible paintings are there? Can I work this out using a simple formula (if such a thing exists?!) Is it thousands?!

(If it's easier to explain with a smaller number maybe start with 4 sections!)

Thanks.

2. Mar 10, 2015

### phyzguy

Let's start by assuming you just have one color, so each section is either colored or not. In your simple case with four sections, it would work out as follows:

(A) No section colored - 1 possibility.
(B) 1 Section colored - 4 possibilities, the colored section is either 1,2,3, or 4
(C) 2 sections colored - 6 possibilities, the colored sections are either 12, 13, 14, 23, 24, or 34.
(D) 3 sections colored - 4 possibilities, the colored sections are either 123, 124, 134, or 234.
(E) 4 sections colored - 1 possibility - 1234.

If you add these up, there are 1+4+6+4+1 = 16 possibilities for 4 sections. It turns out this is a general rule. If there are N sections and n colored sections, there are $\frac{N!}{N! (N-n)!}$ possibilities for coloring them, and when you add them all up, there are 2N possibilities in all. So for your 16 sections, there are 216 = 65,536 possibilities.

3. Mar 11, 2015

### Laura72

Thank you so much for your reply. Yes, I was basing it on just having one color.

I understood the beginning of your answer, but I’m afraid I got lost once you put in the formula!

Would you mind explaining what does ‘N!N!(N−n)!’mean? Would it be possible to write out it in words?

Also, 216 means 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 right?!

Thank you.

4. Mar 11, 2015

### phinds

The exclamation point is the math symbol for "factorial", which means multiply all the integers from 1 to your number. So 3! is 1*2*3 = 6 and 4! is 1*2*3*4 = 24 and so forth.

5. Mar 11, 2015

Right!