Hi Kelley,

You have done most of the work and you have them in a nice order. I am going to call them strings of characters since combination is a word in mathematics that has a technical meaning.

The possible one character strings as you say are 0, 1 and 2. Hence there are 3 possible one character strings.

Each of these you can extend to a two character string by placing either 0, 1 or 2 at the end of the string. Thus 0 extends to 00 01 and 02; 1 extends to 10, 11 and 12 and 2 extends to 20, 21, and 22. Hence you have $3 \times 3 = 3^2$ possible two character strings.

Again each of these you can extend to a three character string by placing either 0, 1 or 2 at the end of the string. Thus 00 extends to 000, 001, and 002; 01 extends to 010, 011 and 012; 02 extends to 020, 021 and 022; and so on. Hence each of the two character strings can be extended to 3, three character strings and there are therefore $3 \times 3 \times 3 = 3^{3}$ possible three digit strings.

Continuing you can see that for any positive integer $n$ there are $3^{n}$ possible n character strings.

I hope this helps,
Penny