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Of the three-digit integers up to 700, how many have two identical dig

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filipembribeiro
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Of the three-digit integers up to 700, how many have two identical dig [#permalink] New post   19 Jul 2021, 04:36
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Of the three-digit integers up to 700, how many have two identical digits and one unique digit?

(A) 55
(B) 108
(C) 109
(D) 162
(E) 163
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E
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chetan2u
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Re: Of the three-digit integers up to 700, how many have two identical dig [#permalink] New post   19 Jul 2021, 05:10
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filipembribeiro wrote:
Of the three-digit integers up to 700, how many have two identical digits and one unique digit?

(A) 55
(B) 108
(C) 109
(D) 162
(E) 163


Number = abc
Let us talk of the first digit, a, as it has restriction of being less than 7.
1) Let the unique digit be a, so a can be any of 6 values from 1 to 6, and other two are identical that can be any of remaining 9 except the value of a.
=> 6*9*1=54.
Add one for 700, so 55.
2) Let a and b be identical
So c can be any of the remaining 9 => 6*1*9=54
3) Let a and c be identical
So b can be any of the remaining 9 => 6*9*1=54

Total 55+54+54=163


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Re: Of the three-digit integers up to 700, how many have two identical dig [#permalink] New post   19 Jul 2021, 06:37
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From 100 through 199, there will be 9 numbers like 1XX, 9 numbers like 11X, and 9 numbers like 1X1 (9 of each since X must be different from 1). So from 100 through 199, there are 27 numbers that meet the conditions. But the same will be true for each range of a hundred numbers, and we have six complete ranges like that from 100 through 699, giving us 6*27 = 162 numbers. We're also asked to include 700, so the answer is 163.
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Re: Of the three-digit integers up to 700, how many have two identical dig [#permalink] New post   25 Mar 2024, 11:37
filipembribeiro wrote:
Of the three-digit integers up to 700, how many have two identical digits and one unique digit?

(A) 55
(B) 108
(C) 109
(D) 162
(E) 163


To start with, remove 700 from it. Now we have to only consider numbers from 100 till 699. Any selection you now make will be in 6C1x9C1 (because one digit is repeating). Let’s denote same as S and different as D. You will have SSD or SDS or DSS either way you have to multiply 6C1x9C1 with 3!/2!

You get 162, but remember we excluded 700 from it. Now 700 has two same digits and one unique digit. Hence, 163 or answer E

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Re: Of the three-digit integers up to 700, how many have two identical dig [#permalink]
25 Mar 2024, 11:37
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