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1.  Kai cuts out all the page numbers of a book. He takes each number and cuts the numbers into separate digits (single digit numbers are already separate). Given that the book has 999 pages numbered 1 through 999, how many separate digits does Kai have when he is finished?

2.  Kai cuts out all the page numbers of a book. He takes each number that has more than one digit and cuts the numbers into separate digits. Given that the book has 999 pages numbered 1 through 999, if he chooses one of these digits at random, what is the probability it is a 7

3.  Kai cuts out all the page numbers of a book. He takes each number that has more than one digit and cuts the numbers into separate digits. Given that the book has 999 pages numbered 1 through 999, what is the sum of all of these digits?

Guest Apr 3, 2015

Best Answer 

 #1
avatar+17711 
+5

How many digits?

From the numbers from 1 through 9, there are 9 digits.

From the numbers from 10 through 99, there are 90 numbers of 2 digits each = 180 digits.

From the numbers from 100 through 999, there are 900 numbers of 3 digits each = 2700 digits.

Total number of digits:  9 + 180 + 2700  =  2889 digits.

How many 7s?

From 000 to 099 there are 10 7s in the one's place and 10 7s in the ten's place = 20 sevens.

There will be 20 sevens in each of the regions 100-199, 200-299, ... 900-999 = 20 x 9 = 180 sevens.

But, in the region 700 - 799, there are 100 7s in the hundred's place.

Total of 300 7s. 

So, the probability of getting a 7:  300/2889

If all the digits from 000 through 999 were used (this includes more zeros than actually printed but won't change the sum), there would be 300 1s, 300 2s, 300 3s, ..., 300 9s.

300(0) + 300(1) + 300(2) + 300(3) + 300(4) + 300(5) + 300(6) + 300(7) + 300(8) + 300(9)  =  13,500

geno3141  Apr 4, 2015
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1+0 Answers

 #1
avatar+17711 
+5
Best Answer

How many digits?

From the numbers from 1 through 9, there are 9 digits.

From the numbers from 10 through 99, there are 90 numbers of 2 digits each = 180 digits.

From the numbers from 100 through 999, there are 900 numbers of 3 digits each = 2700 digits.

Total number of digits:  9 + 180 + 2700  =  2889 digits.

How many 7s?

From 000 to 099 there are 10 7s in the one's place and 10 7s in the ten's place = 20 sevens.

There will be 20 sevens in each of the regions 100-199, 200-299, ... 900-999 = 20 x 9 = 180 sevens.

But, in the region 700 - 799, there are 100 7s in the hundred's place.

Total of 300 7s. 

So, the probability of getting a 7:  300/2889

If all the digits from 000 through 999 were used (this includes more zeros than actually printed but won't change the sum), there would be 300 1s, 300 2s, 300 3s, ..., 300 9s.

300(0) + 300(1) + 300(2) + 300(3) + 300(4) + 300(5) + 300(6) + 300(7) + 300(8) + 300(9)  =  13,500

geno3141  Apr 4, 2015

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