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How to calculate the possible number of combinations

  1. Mar 10, 2015 #1
    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!)

  2. jcsd
  3. Mar 10, 2015 #2


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    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 [itex]\frac{N!}{N! (N-n)!}[/itex] 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.
  4. Mar 11, 2015 #3
    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.
  5. Mar 11, 2015 #4


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    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.
  6. Mar 11, 2015 #5


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