Math CentralQuandaries & Queries

Question from Robin:

I am wondering if you can explain in layman terms how to figure out how many different ways the numbers 00-99 can be arranged? The discussion that sparked this question revolved around the practice of retiring athletes numbers and how many actual number combinations there are. So if I were to use 00, 01, 02 03, 04, 05, 06, 07, 08, 09, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ....99. I say there are only 109 different numbers. I'm being told I'm wrong that there are way more than that. Please help. Thank you, Robin

Hi Robin,

Your answer is correct and I think you have the easiest explanation. You have listed all the possibilities. I will however show you a different approach as sometimes it is not easy to list all the possibilities.

Suppose the number on the jersey has two digits and you are going to put a number on a jersey. You can select the leftmost digit in 10 ways, it can be 0, 1, 2,..., or 9. Once you have selected the leftmost digit and sewed it on you can select the next digit also in 10 ways. Thus you have $10 \times 10 = 100$ ways to sew a two digit number on the jersey. What about 1 digit numbers? You say there are 9 which are 1, 2, 3, ..., and 9. Thus you have in total 109 possible athlete numbers.

My one question is why can't an athlete have the number 0. If she can ten there are 110 possible numbers.



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