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# How many different three-digit numbers can be formed which contain two

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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How many different three-digit numbers can be formed which contain two  [#permalink]

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29 Mar 2018, 01:13
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4
00:00

Difficulty:

55% (hard)

Question Stats:

61% (02:22) correct 39% (02:48) wrong based on 88 sessions

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[GMAT math practice question]

How many different three-digit numbers can be formed which contain two digits that are the same, and a third digit that is different from the other two?

A. 196
B. 216
C. 243
D. 256
E. 316

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Posts: 236
Concentration: Strategy, Finance
GMAT 1: 620 Q46 V29
Re: How many different three-digit numbers can be formed which contain two  [#permalink]

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29 Mar 2018, 02:00
1
Not sure, but should be: 9*9*1*3!/2!=243
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Joined: 08 Jun 2016
Posts: 10
Re: How many different three-digit numbers can be formed which contain two  [#permalink]

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29 Mar 2018, 03:04
The answer should be (B). 9*8*1*3!/2! = 216

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Intern
Joined: 28 Dec 2017
Posts: 5
Re: How many different three-digit numbers can be formed which contain two  [#permalink]

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31 Mar 2018, 06:01
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MathRevolution wrote:
[GMAT math practice question]

How many different three-digit numbers can be formed which contain two digits that are the same, and a third digit that is different from the other two?

A. 196
B. 216
C. 243
D. 256
E. 316

Start from total of ways to obtain 3 digit numbers : 9 * 10 * 10 = 900

The number of ways we are looking for is :Exactly 2 identical digits and 1 different digit (respecting scope constraint : 0 cannot be first digit)

So let's subtract from 900, cases where all digits are different, and cases where all 3 digits are the same, leaving logically the number of ways we are looking for : 900 - (9 * 9 * 8) - (9 * 1 * 1) = 243

Hope it helps
Intern
Joined: 22 Mar 2018
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WE: Operations (Other)
Re: How many different three-digit numbers can be formed which contain two  [#permalink]

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31 Mar 2018, 09:34
2
We can see that there can be 3 scenarios:

1. XXY
2. XYX
3. YXX

XX can be given a value between 1-9 in scenarios 1 and 2. and Y can be given any value 0-9 that is not X so we're looking at 9*9=81

Each scenario there can only be 9*9 outcomes. So 3*81=243. C
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Re: How many different three-digit numbers can be formed which contain two  [#permalink]

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01 Apr 2018, 18:46
=>

These three-digit numbers can have one of the forms XXY, XYX and YXX.
Note that 0 cannot be the hundreds digit.

Case 1): XXY
There are 9 possibilities for X (X is not 0), and 9 possibilities for Y (Y≠X).
This gives a total of 9*9 = 81 possible three-digit numbers of this form.

Case 2): XYX
There are 9 possibilities for X (X is not 0), and 9 possibilities for Y (Y≠X).
This gives a total of 9*9 = 81 possible three-digit numbers of this form.

Case 3): YXX
There are 9 possibilities for Y (Y is not 0), and 9 possibilities for X (X≠Y).
This gives a total of 9*9 = 81 possible three-digit numbers of this form.

Thus, the total number of possible three-digit numbers is 81 + 81 + 81 = 243.

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Re: How many different three-digit numbers can be formed which contain two  [#permalink]

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19 Dec 2018, 10:44
Hi Terrifficm

suppose the two same digits are x and one different digit is y

Answer is not 216 because, for two same digits we can select from nine digits - 1,2,3,4,5,6,7,8,9. Therefore, it would be 9C1 i.e. 9

and for one different digit , we can select from - 0,1,2,3,4,5,6,7,8,9 i.e 10-1 digit i,e again 9 digits. therefore, it would be 9C1 i.e.9

Again these three digits can be arranged among themselves in three ways for e.g. xxy, xyx, yxx

thus the calculation is 9*9*3 which is 243

trust this helps
Re: How many different three-digit numbers can be formed which contain two   [#permalink] 19 Dec 2018, 10:44
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