I have 3 different colors. How many combinations of 3 are possible if color and position can be repeated?

Question

I'm not quite sure what to put in this space?

Answer 1

Hello Gregg,

 

First let's pick 3 colors: red (r), blue (b), yellow (y)

 

Since any color can be repeated, that means the number of combinations can be see like this:

 

Each spot can be any one color (rrr, bbb, or yyy).  There are only 3 combinations for this.

Two spots can be one color, combined with a second (rrb, rry, ryr.....). There are 18 combinations for this arrangement (3x3x2).

Finally, each spot can be a different color (rby, ryb, bry.....). There are 6 combinations for this arrangement (3x2x1).

 

In total, this means there are 27 different combinations.  They are listed below:

 

rrr      bbb     yyy

rrb      bbr     yyr

rbr      brb     yry

brr      rbb     ryy

rry      bby    yyb

ryr      byb    yby

yrr      ybb    byy

rby      bry    yrb

ryb      byr    ybr

 

Hope this helps!

Answer 2

Daniel W. and Jayden H. have written good answers, but here is a combination math-reasoning approach:

 

Let's start with one Letter Position which could be any of three colors (we'll use R-G-B because I've seen that with computer graphics).  How many unique values could we have?   ---  just 3.

 

O.K. What about with two Letter Positions?  Now, for each selection of the first letter, there are 3 options for the second letter.  That's 3*3.    (note: in decimal, from 00 to 99 is 10*10=100 possible values, right?).

 

Then, with 3 Letter Positions?   3*3*3=27 unique possibilities.

 

This number is small enough to enumerate the possibilities to help your understanding (like the other tutors did), but the   digits^base  expression (with "^" meaning exponentiation) is important.

Answer 3

You have three different colors, lets say red (R), green (G) and blue (B) and you choose 3 colors. The colors can repeat. The following can be mathematically represented as 3³ = 27.

 

RRR
RRB
RRG

RBR
RBB
RBG

RGR
RGB
RGG

 

BRR
BRB
BRG

BBR
BBB
BBG

BGR
BGB
BGG

 

GRR
GRB
GRG

GBR
GBB
GBG

GGR
GGB
GGG