A 3-digit positive integer consists of non zero digits. If

This topic has expert replies
Moderator
Posts: 2021
Joined: 15 Oct 2017
Followed by:6 members

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

Source: GMAT Prep

A 3-digit positive integer consists of non zero digits. If each exactly two of the digits are the same, how many such integers are possible?

A. 72
B. 144
C. 216
D. 283
E. 300

The OA is C

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 2094
Joined: 04 Dec 2012
Thanked: 1443 times
Followed by:247 members

by ceilidh.erickson » Thu May 16, 2019 12:08 pm
There are 3 configurations that would yield a 3-digit integer with 2 of the same digit, one different digit:
[same][same][different]
[same][different][same]
[different][same][same]

So, let's calculate the number of combinations for one of these:
[same][same][different]
There would be 9 non-zero digits as options for the hundreds digit, and then only 1 option for the tens digit, since it has to be the same as the hundreds digit. You might think that there would be 9 options for the units digit, but remember - it has to be different! So there would be only 8 options that are different from whatever we picked for the hundreds & tens digits.
9*1*8 = 72 options

It would stand to reason that all 3 configurations mentioned above would have the same number of possibilities: 72. So we add:
72 + 72 + 72 = 216

The answer is C.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

User avatar
GMAT Instructor
Posts: 15533
Joined: 25 May 2010
Location: New York, NY
Thanked: 13060 times
Followed by:1901 members
GMAT Score:790

by GMATGuruNY » Thu May 16, 2019 1:55 pm
BTGmoderatorLU wrote:Source: GMAT Prep

A 3-digit positive integer consists of non zero digits. If each exactly two of the digits are the same, how many such integers are possible?

A. 72
B. 144
C. 216
D. 283
E. 300
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:
Number of options for the hundreds digit = 9. (Any digit but 0.)
Number of options for the tens digit = 9. (Any digit but 0.)
Number of options for the units digit = 9. (Any digit but 0.)
To combined these options, we multiply:
9*9*9.

Integers with all 3 digits the same:
111, 222, 333, 444, 555, 666, 777, 888, 999.
Number of options = 9.

Integers with all 3 digits different:
Number of options for the hundreds digit = 9. (Any digit but 0.)
Number of options for the tens digit = 8. (Any digit 1-9 other than the digit already used.)
Number of options for the units digit = 7. (Any digit 1-9 other than the two digits already used.)
To combine these options, we multiply:
9*8*7.

Thus:
Integers with exactly 2 digits the same = (9*9*9) - 9 - (9*8*7) = 9(81-1-56) = 9(24) = 216.

The correct answer is C.
Mitch Hunt
Private Tutor for the GMAT and GRE
[email protected]

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
For more information, please email me at [email protected].
Student Review #1
Student Review #2
Student Review #3

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 6365
Joined: 25 Apr 2015
Location: Los Angeles, CA
Thanked: 43 times
Followed by:26 members

by [email protected] » Thu May 16, 2019 6:52 pm
BTGmoderatorLU wrote:Source: GMAT Prep

A 3-digit positive integer consists of non zero digits. If each exactly two of the digits are the same, how many such integers are possible?

A. 72
B. 144
C. 216
D. 283
E. 300

The OA is C
The 3 digits are nonzero, and they consist of two distinct digits. The number of ways to choose 2 digits from 9 nonzero digits is 9C2 = (9 x 8)/2 = 36. Now for each pair of digits chosen, for example, 1 and 2, we could have: 112, 121, 211, 221, 212, and 122. Therefore, there are 36 x 6 = 216 such integers.

Answer: C

Scott Woodbury-Stewart
Founder and CEO
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

ImageImage