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integers

by sgr21 » Wed Apr 16, 2014 9:36 am
Of the three-digit integers greaterthan 700, how many
have two digits that are equal to each other and the
remaining digit different from the other two?

(A) 90
(B) 82
(C) 80
(D) 45
(E) 36

P.S. please can anyone give me a brief intro about integers as this is the first time i am studying gmat and also because i have lost touch of math.

thnk u
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by Brent@GMATPrepNow » Wed Apr 16, 2014 9:44 am
sgr21 wrote:Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

(A) 90
(B) 82
(C) 80
(D) 45
(E) 36
One approach is to start LISTING numbers and look for a PATTERN.

Let's first focus on the numbers from 800 to 899 inclusive.
We have 3 cases to consider: 8XX, 8X8, and 88X

8XX
800
811
822
.
.
.
899
Since we cannot include 888 in this list, there are 9 numbers in the form 8XX

8X8
808
818
828
.
.
.
898
Since we cannot include 888 in this list, there are 9 numbers in the form 8X8

88X
880
881
882
.
.
.
889
Since we cannot include 888 in this list, there are 9 numbers in the form 88X

So, there are 27 (9+9+9) numbers from 800 to 899 inclusive that meet the given criteria.

Using the same logic, we can see that there are 27 numbers from 900 to 999 inclusive that meet the given criteria.

And there are 27 numbers from 700 to 799 inclusive that meet the given criteria. HOWEVER, the question says that we're looking at numbers greater than 700, so the number 700 does not meet the criteria. So, there are actually 26 numbers from 701 to 799 inclusive that meet the given criteria.

So, our answer is 27+27+26 = [spoiler]80 = C[/spoiler]

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Sun Apr 05, 2015 6:57 am, edited 1 time in total.
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by Brent@GMATPrepNow » Wed Apr 16, 2014 9:48 am
sgr21 wrote: P.S. please can anyone give me a brief intro about integers as this is the first time i am studying gmat and also because i have lost touch of math.
Integers can be thought of a "whole" numbers.
55, 19, -5 and 0 are examples of integers.
22.7, 1/2, and -333.1 are examples of non-integers.

The question refers to 3-digit integers greater than 700.
So, we need only consider the integers 701, 702, 703, ....., 998, and 999

Cheers,
Brent
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by GMATGuruNY » Wed Apr 16, 2014 12:11 pm
Of the 3 digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other two?

A. 90
B. 82
C. 80
D. 45
E. 36
An alternate approach:

Integers with exactly 2 digits the same = Total integers - Integers with all 3 digits the same - Integers with all 3 digits different.

Total integers:
To count consecutive integers, use the following formula:
Number of integers = biggest - smallest + 1.
Thus:
Total = 999 - 701 + 1 = 299.

Integers with all 3 digits the same:
777, 888, 999.
Number of options = 3.

Integers with all 3 digits different:
Number of options for the hundreds digit = 3. (7, 8, or 9)
Number of options for the tens digit = 9. (Any digit 0-9 other than the digit already used.)
Number of options for the units digit = 8. (Any digit 0-9 other than the two digits already used.)
To combine these options, we multiply:
3*9*8 = 216.

Thus:
Integers with exactly 2 digits the same = 299-3-216 = 80.

The correct answer is C.
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