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Re: How many integers between 1 and 10^21 [#permalink]
15 Oct 2010, 23:55

Pkit wrote:

How many integers between 1 and 10^21 (10 in 21st power) are such that the sum of their digits is 2?

a.190 b.210 c.211 d.230 e.231

Bunuel, please leave a chance for other people to solve this.

thanks

between 1 and 10^21 means we are talking about numbers with upto 21 digits.

Case 1 Numbers of the form 2,20,200,etc Possible numbers = 21 (1 possibility for each number of digits)

Case 2 Numbers using the digits {1,1} Such a number will have a minimum of 2 digits and a maximum of 21 digits, with the first digit=1. Now, consider the k-digit number, with the first digit "1", out of the rest of the k-1 digits, we have k-1 ways to choose where to put the second 1. So k-1, k-digit numbers possible. So total number of numbers = Summation(k-1;k=2 to k=21)=1+2+...+20=20(21)/2=210

Re: How many integers between 1 and 10^21 [#permalink]
16 Oct 2010, 02:15

so the second condition means there are 21 positions and two number, 1 and 1 so place in then. hence we can use the combination formula 21c11 and get 210.

Re: How many integers between 1 and 10^21 [#permalink]
16 Oct 2010, 04:33

Wow!! whenever I see these kinda questions I got stunned but guys like bunuel,shrouded1, and gurpreet are there to help . Nice method gurpreet and whenever a quant question is posted, shrouded1 is always there to explain it. Thank you! guys (I couldn't solve it but now i know how to solve it ) _________________

"Don't be afraid of the space between your dreams and reality. If you can dream it, you can make it so." Target=780 http://challengemba.blogspot.com Kudos??

How many integers between 1 and 10^21 are such that the sum [#permalink]
04 Dec 2012, 13:47

3

This post received KUDOS

Expert's post

How many integers between 1 and 10^21 are such that the sum of their digits is 2? (A) 190 (B) 210 (C) 211 (D) 230 (E) 231

For a complete explanation, see: http://gmat.magoosh.com/questions/835 When you submit your answer, the next page will have a complete video explanation. Each one of Magoosh's 800+ practice GMAT questions has its own video explanation, for accelerated learning. _________________

Re: How many integers between 1 and 10^21 are such that the sum [#permalink]
28 Dec 2012, 09:10

Combinations of digits that sum up to 2 are either 1+1 or 2+0:

Combinations 1+1:

Double Digits 11 Total: 1

Triple Digits 101 110 Total: 2

Quadruple Digits 1001 1010 1100 Total: 3

So now the pattern emerges that for every N digit number, there are N-1 combinations summing up to two. The number of integers should be between 1 and 10^21, exclusive which means that the largest allowed integer does only have 21 digits, not 22 which means that the amount combinations with the largest amount of places is 20 (21-1).

We can thus calculate the total amount of integers that satisfy the question by calculating the set of consecutive integers: 1+2+3...+20 which is 210. However, we did not account for the combinations of 2 and 0 that sum up to two, we have to add them first: