How many combinations of 3 variables are possible when single variables can't be repeated

Question

This is a bit of a strange and specific question, but I've been putting a lot of thought into this problem and can only find 8 totally unique combinations, while I thought there'd be 9. I hope I worded this well, lol.

 

Here are the combinations I can think of, with 0 representing no variables.

1      2      3
12    13    23
0      123

 

Are there only 8 instead of 9? Am I looking at this the wrong way? Any help would be appreciated, thank you.

Answer 1

The total number of combinations of n things is always 2ⁿ, that is 8 in your case and you have listed them correctly.

 

Remember that then sum of _nC_r from r = 0 to n is 2ⁿ, which you can prove by expanding (1 + x)ⁿ by the binomial theorem and then setting x = 1...and if you are really unclear you can prove the expansion by induction!

 

There is another way of seeing this and I can't remember...but check out "Choice and Chance" by Whitworth, published in the late 1800s but still a book referred to by actuaries and others.