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

Question

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

chetan2u · Answer

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

E

IanStewart · Answer

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.

chetanraghav17 · Answer

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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Dereno · Answer

Let the three digit number be ABC
Lets find the number of distinct three number digits from 100 to 700, 
A can hold values from 1 to 6,
B can hold values between 0 to 9, but leaving the digit chosen for A. So left with 10-1 = 9 options. 
C can hold values apart from A and B , so 8 options
6*9*8 = 432 values with unique distinct digits.

Let’s look into three digits within 600, having all digits same 111,222,333,444,555,666 = total 6 values.
Hence 432+6 = 438.

Now to the question two identical and one unique digits between 100 to 700 (600 values).
600 - 438 = 162

We haven’t considered 700 during the initial process. Hence, 162+1 = 163

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

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

Prep Toolkit

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