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# How many integers between 1 and 10^21 are such that the sum

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Senior Manager
Joined: 20 May 2014
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How many integers between 1 and 10^21 are such that the sum [#permalink]  05 Oct 2017, 20:35
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Question Stats:

32% (01:54) correct 67% (00:56) wrong based on 28 sessions
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

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[Reveal] Spoiler: OA
Director
Joined: 20 Apr 2016
Posts: 761
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Kudos [?]: 517 [1] , given: 94

Re: How many integers between 1 and 10^21 are such that the sum [#permalink]  05 Oct 2017, 21:22
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Bunuel wrote:
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

Kudos for correct solution.

10^21 here we have two option

1. 1+1 = 2

2. 2+0 =2

Now for the first option ,

10^21 can be written as = 1,000,000,000,000,000,000,000 i.e 21 zero's

Here we have to fill two places with 1 out of 21 places i.e 11,1001,100001 ....(since the number has to be less than 10^21)

so we can use the combination formula as 21C2 as we are choosing two places from the 21 places of zero's
=210 ways.

Now for the option 2, choosing 2
we have 21 ways i.e 02,020,200,2000 ....(i.e we can fill up the 21 zero's places with 2,as the number has to be less than 10^21)

Therefore total no of ways =210+21=231 ways.
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Intern
Joined: 22 Jul 2018
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Kudos [?]: 9 [0], given: 5

Similar problems [#permalink]  14 Oct 2018, 01:27
1. How many integers between 2,000 and 3,999 have a ones digit??
2. How many integers between 0 and 1570 have a prime tens digit and a prime units digit?
Director
Joined: 20 Apr 2016
Posts: 761
Followers: 6

Kudos [?]: 517 [0], given: 94

Re: Similar problems [#permalink]  15 Oct 2018, 19:44
ruposh6240 wrote:
1. How many integers between 2,000 and 3,999 have a ones digit??
2. How many integers between 0 and 1570 have a prime tens digit and a prime units digit?

The first ques- is not complete

2. there are 252 integers.

Next time would request you to read the rules before posting, as this type of post are not encouraged

https://greprepclub.com/forum/rules-for ... -1083.html
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Supreme Moderator
Joined: 01 Nov 2017
Posts: 213
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Kudos [?]: 70 [0], given: 1

Re: How many integers between 1 and 10^21 are such that the sum [#permalink]  15 Oct 2018, 22:15
Expert's post
Bunuel wrote:
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

Kudos for correct solution.

How many digits will 10^n have ;- n+1, so 10^21 will have 22 digits - 1 of one and 21 of zeroes
but we are looking for < 10^21, so we will have 22-1=21 digits to fill up..

we can get 2 in two forms
1) 1+1
2) 2+0

so
1) 1+1..... choose 2 places out of 21 for two 1s....21C2=21*10=210
2) 2+0...... choose one place out of 21 for one 2.... 21C1=21

total = 210+21=231
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Intern
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Re: How many integers between 1 and 10^21 are such that the sum [#permalink]  03 Nov 2018, 06:37
Intern
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Re: How many integers between 1 and 10^21 are such that the sum [#permalink]  04 Nov 2018, 22:38
231 ways. 21C2+ 21C1
Re: How many integers between 1 and 10^21 are such that the sum   [#permalink] 04 Nov 2018, 22:38
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