Math CentralQuandaries & Queries


Question from Jon:

I'm not a mathematician but just curious. Is there a name for a number of up to ten digits where each digit is different, i.e the equivalent of an isogram in linguistics? My online banking gives me a random 8-digit number each time I log on and it is rare to get one of the type I refer to. There must be calculable odds, but I'm only allowed one question!
Thanks in advance

Hi Jon,

I am not aware of any word for the type of number you are describing, one where all the digits are different. I was surprised in your question by the word isogram. The prefix iso has a Greek root and means equal or having equal value so it seems a strange prefix for a term to describes a word where all the letters are different. I did find this meaning for isogram a few places on the web but I wasn't able to find its etymology. I decided to look in the Oxford English Dictionary which gave the meaning of the word isogram as:

A proposed general term for lines on a diagram, etc. indicating equality of some physical condition or quantity, as isotherms, isobars, etc.

This seems to me as a more natural meaning of isogram but that wasn't your question.

As for your bank can the 8 digit random number have its leading digit as zero? For example is 07569232 allowed? If so there are 10 choices for the first digit, 10 choices for the second digit and so on so the bank's random number generator is choosing form

\[10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^8\]

possibilities. If an 8 digit "isogram" can also start with a zero then there are 10 choices for the first digit, 9 choices for the second digit, 8 choices for the third digit and so on. This gives

\[10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3\]

8 digit "isograms". Hence the probability that the bank will present you with an "isogram" is

\[\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{10^8}\]

I had Google do this calculation for me. I typed (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3)/(10^8) into the Google search window and Google responded with 0.018144 or a little less than two in a hundred.



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