When a telephone exchange is established, there are 10,000 new phone numbersto use(i.e. 722-0000 to 722-9999). Of these 10,000 phone numbers, how many do not contain the digit "9"?

I know I can do it this long way, but how do I do it in a shorter more reliable way.

722-0000
722-0001
722-0002
722-0003

Ect....
Thank you

with no restrictions, number of 4 digit permutations are

10x10x10x10 = 10,000 as you stated

but , if we exclude the 9, we have only 9 numerals to play with

number of 4 digit phone number endings
= 9x9x9x9 = 6561

(now, a nicer question would have been:
of those 4 digit endings, how many would contain a pair of 9's)

Thank you Reiny.

That makes perfect sense to me. Next time I post a question I will think a little harder to possible ways of working questions like this.
😀😀

To find the number of phone numbers that do not contain the digit "9" in the given range (722-0000 to 722-9999), you can use a simpler and more efficient approach.

First, understand that each of the four digits in the phone number has 10 possibilities (0-9), except for the first digit, which cannot be 0. Therefore, there are 10 options for the first digit (7), and 10 options each for the remaining three digits.

To calculate the total number of phone numbers in the range, you multiply these possibilities together: 10 × 10 × 10 × 10 = 10,000.

Since you want to exclude the numbers containing the digit "9," you need to determine how many phone numbers do contain it.

To count the number of possibilities for each digit, subtract 1 (for excluding the digit "9") from the total number of possibilities, which is 10.

Therefore, the number of phone numbers that do not contain the digit "9" is:
9 × 10 × 10 × 10 = 9,000.

So, out of the 10,000 phone numbers in the range, 9,000 do not contain the digit "9."