Math CentralQuandaries & Queries


Question from Lulamile:

What is the probability that a six-digit telephone number has no repeated digits? The telephone number cannot begin with a zero


For the possible phone numbers there are 9 choices for the first digit and 10 choices for each of the remaining 5 digits. Thus there are $9\times 10^5$ possible phone numbers.

How many of them meet the criteria of having no repeated digits? Again there are 9 choices for the first digit. Once it has been chosen there are only 9 choices for the second digit since the first digit can't be repeated. Thus you have $9 \times 9$ choices for the first two digits. For the third digit you have 8 choices since you can't repeat either of the two you have already used. Thus you have $9 \times 9 \times 8$ choices for the first three digits.

You can do the rest.


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