Quandaries
and Queries 

Help. My name is John, 12th and the teacher wants us to tell him the answer to the following math query. With a nine digit number, with each of the nine digits having a possibility of 10 different numbers then what is the total number of possible mathematical variations in the nine digit number. i.e. Social Security numbers have nine digits and if each of the nine digits have a possibility of being any one of ten numbers, i.e. 0,1,2,3,4,5,6,7,8,9. Then what is the formula to calculate the maximum possible number of variations in this nine digit number and what is the mathematical maximum possible number of variations of this nine digit number? The answer? 

Hi John, Let's solve a smaller problem. Suppose you want to make a four digit number rather than a 9 digit number. Suppose also that you can use only 0,1,2 rather than 0,1,2,3,4,5,6,7,8,9. So you might have numbers like 0210, or 2120, or ... These numbers are so small that we might be able to list them all if we do it in an orderly fashion. Start by writing down all the possibilities for the first digit. They are
Use each of these as a base for a two digit number. From each of the one digit numbers you can construct 3, two digit number, they are
Hence there are 3 3 = 9, two digit numbers. Use each of these as a base for a three digit number. From each of the two digit numbers you can construct 3, three digit number, they are
Hence there are 3 3 3 = 27, three digit numbers. Use each of these as a base for a four digit number. From each of the three digit numbers you can construct 3, four digit number, they are
I am not going to list them, but you can see that there are 3 3 3 3 = 81, four digit numbers. Can you see how to modify this construction to count the number of nine digit numbers you can construct from 0,1,2,3,4,5,6,7,8,9? Penny 

