Restriction

[jain2016]

How many 3-digit integers greater than 499 have exactly two repeated digits? ( e.g. , 606, 500, 772...)

A) 90

B) 105

C) 108

D) 135

E) 140

OAD

Hi Experts ,

We have a restriction in this question, so can we solve this question by restriction formula i.e.

# of ways to follow restriction = # of ways to ignore restriction - # of ways to break restrictions.



This question is from GMATPREPNOW video.

Please explain.

Many thanks in advance.

SJ

[GMATGuruNY]

How many 3-digit integers greater than 499 have exactly two repeated digits? ( e.g. , 606, 500, 772...)

A) 90

B) 105

C) 108

D) 135

E) 140

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 - 500 + 1 = 500.

Integers with all 3 digits the same:
555, 666, 777, 888, 999.
Number of options = 5.

Integers with all 3 digits different:
Number of options for the hundreds digit = 5. (5, 6, 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:
5*9*8 = 360.

Thus:
Integers with exactly 2 digits the same = 500-5-360 = 135.

The correct answer is D.

[Brent@GMATPrepNow]

Here's the video solution (which is different from Mitch's excellent solution): https://www.gmatprepnow.com/module/gmat ... /video/796

Cheers,
Brent

[Matt@VeritasPrep]

You could also think of it in the way you suggest, by considering numbers for which all three digits are different. There are 5 options for the hundreds digit, 9 for the tens digit (the 8 we didn't use for the hundreds, plus 0), and 8 for the units digit (the 8 we haven't used so far), making 5 * 9 * 8 numbers.

We also have to consider the three digit multiples of 111 (555, 666, 777, 888, 999), as those don't count either.

From here, we do

Total - (all digits different) - (all three digits same) =

500 - 5*9*8 - 5 =

135