How to calculate the possible number of combinations

[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.

 

[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 2^N possibilities in all. So for your 16 sections, there are 2¹⁶ = 65,536 possibilities.

 

[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.

 

[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.

 

[phyzguy]

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?!

Right!