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avatar+183 

How many 5-digit numbers have at least one 2 or one 3 among their digits?

 Oct 18, 2018
 #1
avatar+98090 
+2

There are  99,999 - 10,000 + 1  = 90,000 five digit numbers

 

This equals   90,000  less the  integers that contain no 2's or 3's

 

We have  7 ways to pick the leading digit  [ since 0 cannot lead ] and 8 ways to pick  each of the other 4 digits

 

So      7 * 8^4  =  28,762

 

So   90,000 - 28,762  =  61328  integers

 

 

cool cool cool

 Oct 18, 2018
 #2
avatar+183 
+1

Thanks!

 Oct 18, 2018
 #3
avatar
0

Of all the numbers between 10,000 and 99,999 =90,000 numbers, there are 300 2s in every 1,000 numbers. From this 300 will subtract repeats of which there are 10 per 1,000, namely:22, 122, 222, 322, 422, 522, 622, 722, 822, 922. That leaves us with 300 - 10 =290 numbers with 2s in them. So, 290 x 90,000/1,000 =26,100 

 

   In the 4th position, we have 1,000 2s x 9 =9,000 2s. Such as:12,000, 22,000, 32,000, 42,000, 52,000, 62,000, 72,000, 82,000, 92,000.


  In the 5th position, we have 10,000 2s, namely:20,000 to 29,999. So, in total, we have:
26,100 + 9,000 + 10,000 =45,100 numbers with 2s between 10,000 and 99,999. This is in accord with the following computer count of ALL digits between 10,000 and 99,999, which adds 900 repeated(or 10 x 90,000/1,000) 2s to the count:


(0=36 000, 1=46 000, 2=46 000, 3=46 000, 4=46 000, 5=46 000, 6=46 000, 7=46 000, 8=46 000, 9=46 000)
The above applies exactly the same to 3s =45,100.

 Oct 19, 2018

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