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# How many three digit numbers that do not contain the digit 2

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Senior Manager
Joined: 21 Oct 2013
Posts: 414
How many three digit numbers that do not contain the digit 2  [#permalink]

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18 Jul 2014, 11:24
2
8
00:00

Difficulty:

35% (medium)

Question Stats:

65% (01:25) correct 35% (01:36) wrong based on 150 sessions

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How many three digit numbers that do not contain the digit 2 are there?

A. 100
B. 200
C. 512
D. 648
E. 729
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Joined: 02 Sep 2009
Posts: 56244
Re: How many three digit numbers that do not contain the digit 2  [#permalink]

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18 Jul 2014, 11:52
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goodyear2013 wrote:
How many three digit numbers that do not contain the digit 2 are there?

A. 100
B. 200
C. 512
D. 648
E. 729

8*9*9 = 648: 8 options for the first digit (no 0, no 2) and 9 options for each of the remaining digits (no 2 there).

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Re: How many three digit numbers that do not contain the digit 2  [#permalink]

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26 Jan 2015, 12:00
Applying combinations....8 * 9 * 9 = 648...

8 for first digit...cant be 0 or 2
9 and 9 for 2nd and 3rd....cant be 2
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Re: How many three digit numbers that do not contain the digit 2  [#permalink]

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26 Jan 2015, 12:12
Hi All,

This is a great permutation question (and Bunuel's solution uses that type of math). If you don't "see" the permutation approach, there's still another way to get to the correct answer....The answer choices to this question are "spread out" enough that we can actually estimate the correct answer with a bit of logic:

We're asked to figure out how many 3-digit positive integers do NOT contain the number 2.

From 1 to 999, inclusive, there are 999 numbers. 1 through 99 are NOT 3-digit numbers, so we have to remove those.

999 - 99 = 900 3-digit numbers to start.

Now, how many 3-digit numbers "start" with a 2? Every number from 200 to 299 ---> 100 numbers that we have to remove.

Next, how many 3-digit numbers "end" with a 2? One out of every 10 --->900(1/10) ---> Approximately 90 additional numbers that we have to remove. (NOTE: some of these numbers have ALREADY been removed by the prior calculation. The number 202, for example, so we can't remove it twice).

Finally, how many 3-digit numbers have a 2 "in the middle"? This is the same calculation as the prior one. One of of every 10 --> 900(1/10) --> Approximately 90. (NOTE: again, some of these numbers have ALREADY been removed by the prior calculations. The numbers 222 and 225, for example, so we can't remove them again.)

So we're left with approximately 900 - 100 - 90 - 90 = 900 - 280 = approximately 620.

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Re: How many three digit numbers that do not contain the digit 2  [#permalink]

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18 Sep 2018, 16:59
Please explain where my thought process is wrong:

Total 3 digit number - 900

Digits with 2 in hundred's place - 1x10x10
Digits with 2 in ten's place - 9x1x10
Digits with 2 in unit's place - 9x10x1

So if I subtract (900 - 280), i get 620..
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Re: How many three digit numbers that do not contain the digit 2  [#permalink]

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18 Sep 2018, 21:19
Hi PLUTO,

The reason why the answer is not exactly 620 is because some of the 3-digit numbers in your 3 lists "overlap" - meaning that you are counting them 2 or 3 times. For example, consider the numbers 227 and 202 - they both appear in 2 of your lists (and the number 222 appears in all 3 lists). However, each of those numbers is only supposed to be 'counted' once - and if a number appears in 2 lists then you have counted it twice.

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Re: How many three digit numbers that do not contain the digit 2   [#permalink] 18 Sep 2018, 21:19
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