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# How many 3-digit integers can be chosen such that

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How many 3-digit integers can be chosen such that [#permalink]  22 Aug 2017, 02:26
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Question Stats:

41% (01:57) correct 58% (00:55) wrong based on 17 sessions

How many 3-digit integers can be chosen such that none of the digits appear more than twice, and none of the digits equal 0?

(A) 729
(B) 720
(C) 648
(D) 640
(E) 576
[Reveal] Spoiler: OA

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Re: How many 3-digit integers can be chosen such that [#permalink]  25 Aug 2017, 04:16
Can somebody explain? I think I did not get the question.
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Re: How many 3-digit integers can be chosen such that [#permalink]  25 Aug 2017, 06:09
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boxing506 wrote:
Can somebody explain? I think I did not get the question.

The question askes you to select a 3 digit number. So we need to selct 3 digits from {1, 2, 3, 4, 5, 6, 7, 8, 9}.

Now number of ways we can select 1 digits = 9. Since there are 9 options to choose from!
Now number of ways we can select 3 digits = $$9 \times 9 \times 9 = 729$$

Now the above section contains digits like = 999, 888, 777 ..... they are not allowed (3-digit integers with all alike digits )

So correct answer is $$729 - 9 = 720$$. Hence B.
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Last edited by Carcass on 25 Aug 2017, 06:47, edited 1 time in total.
Edited by Carcass
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Re: How many 3-digit integers can be chosen such that [#permalink]  16 May 2018, 09:24
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Carcass wrote:

How many 3-digit integers can be chosen such that none of the digits appear more than twice, and none of the digits equal 0?

(A) 729
(B) 720
(C) 648
(D) 640
(E) 576

Let’s first disregard the condition of none of the digits appearing more than twice and count the three-digit numbers where none of the digits is 0. We see that there are 9 choices for each of the digits; therefore, there are 9^3 = 729 such numbers.

Now, we can deal with the condition that none of the digits should appear more than twice. If a digit of a 3-digit number appears more than twice, then it must appear all 3 times and there are only 9 numbers that have this property: 111, 222, …, 999. Thus, out of the 729 numbers, 9 do not satisfy this property and 729 - 9 = 720 do satisfy.

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Re: How many 3-digit integers can be chosen such that   [#permalink] 16 May 2018, 09:24
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