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# Between 100 and 200, how many numbers are there in which one digit is

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Intern
Joined: 01 Nov 2015
Posts: 20
Between 100 and 200, how many numbers are there in which one digit is  [#permalink]

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08 Jun 2016, 11:55
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95% (hard)

Question Stats:

15% (02:33) correct 85% (02:24) wrong based on 66 sessions

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Between 100 and 200, how many numbers are there in which one digit is the average of the other two?

A) 11
B) 12
C) 10
D) 8
E) 9
Senior Manager
Joined: 23 Apr 2015
Posts: 295
Location: United States
WE: Engineering (Consulting)
Re: Between 100 and 200, how many numbers are there in which one digit is  [#permalink]

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08 Jun 2016, 12:49
1

Let's list the numbers between 100 and 200 that have digits in which one of them is average of the other 2.

{102, 111, 123, 135, 147, 159 } in these five, except 111, the others has 2 ways of rearranging, i.e., 123 can also be written as 132, and so is for 135 as 153.

Therefore 1 + 2 * 5 numbers = 11

Share a Kudo , if you like this explanation
Senior Manager
Joined: 13 Oct 2016
Posts: 363
GPA: 3.98
Between 100 and 200, how many numbers are there in which one digit is  [#permalink]

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16 Dec 2016, 04:42
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aayushagrawal wrote:
Between 100 and 200, how many numbers are there in which one digit is the average of the other two?

A) 11
B) 12
C) 10
D) 8
E) 9

How can we solve this question without actually listing all possibilities?

First of all we need to notice that if we have three digits, one of them will be the average of the others only in case when those other two digits have the same parity.

In other words in ABC B will be the average only if A + C = even number. Now let’s look at our question. We have 1AB.

Case #1

A will be the average. 1(Average)B In order for A to be the average B must be odd because 1 is odd and we have: 1*1*5 = 5 ways.

Case #2

B will be the average. 1A(Average). In this case A must be odd and we have: 1*1*5 = 5 ways

Together 5 + 5 = 10. Right? No, because we have double counting. In each case we can get 111 as a result so we need to subtract one case.

10 – 1 = 9

Case #3

When actual average will be our 1 in front: (Average)AB. Because our average is 1 this can be achieved only in 2 ways:

102 and 120.

Finally our total result will be: 9 + 2 = 11

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Joined: 09 Sep 2013
Posts: 11710
Re: Between 100 and 200, how many numbers are there in which one digit is  [#permalink]

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19 Aug 2018, 09:38
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Between 100 and 200, how many numbers are there in which one digit is   [#permalink] 19 Aug 2018, 09:38
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