Of the three-digit integers greater than 600, how many have

Question

Of the three-digit integers greater than 600, how many have two digits that are equal to each other and the remaining digit different from the other two?

(A) 120
(B) 116
(C) 108
(D) 107
(E) 72

Bunuel · Accepted Answer

# of three-digit integers greater than 600 is 399 (999-600);

Numbers with all distinct digits: 4*9*8=288. First digit can take 4 values: 6, 7, 8 or 9, second digit can take 9 values (10 minus the one we used for the first digit) and third digit can tale 8 values (10 minus 2 digits we've already used for the first two digits);

Numbers greater than 600 which have all alike digits: 4 (666, 777, 888, 999);

{Total}-{all distinct}-{all alike}={two alike, one different} --> 399-288-4=107.

Answer: D.

EvaJager · Answer

The question is about counting/combinations.

The numbers can be of the form  AAB, \,ABA,  or  ABB  where  A  and  B  are different digits and  A\geq6 .
There will be  4\cdot9\cdot3=108  possibilities, but we have to subtract  1  for the number  600  (which is obtained for  A = 6  and  B = 0 ).
Therefore, total number of possibilities  108 - 1= 107.

Answer D.

enigma123 · Answer

Thanks for very clear and precise explanation Bunuel. Highly appreciate.

cminoiu · Answer

If the problem asks for numbers greater than 600, don't we have to start counting from 601 forward? I think we don't, but just double-checking to make sure.... thanks!

EV · Answer

Bunuel, thanks for the explanation. Mine looks more complicated, but could you please check if my solution is correct.

1) yxx - first digit can have 4 values, second - 9, third - 1 = 36
2) yyx - first digit - 4, second - 1, third - 9 = 36
3) yxy - first digit - 4, second 9, third - 1 = 36

1)+2)+3) = 108 - 1*= 107
1* - we should exclude 600

GMATPill · Answer

Ev,

what you did was basically the same as what evajager did.

Let me clarify.

Three digit number: 6xy

First digit: either 6, 7, 8, or 9

So out of 4 possible digits, you choose 1.  (4C1)

Either 2nd or 3rd digit must match the first digit -> Only 1 possibility so  (1C1)

Then the remaining last digit (opposite of the you chose above) can have 9 remaining digits to choose from. So  (9C1) .

Of course, out of 3 available spots, we have to pick 2 of them to be the same  (3C2).

So multiply them together:

= # of ways to arrange a pair within 3 slots *  
= (3C2)

Since the question technically asks for numbers >600, the exact value of 600 (which was included) should not be included.

So we subtract 108 - 1 = 107

EMPOWERgmatRichC · Answer

Hi All,

In these types of questions, the real issue is thoroughness - make sure that you're not "missing" any of the possibilities and make sure that you're not "counting" a possibility that should NOT be counted (or accidentally counting a possibility more than once). Your ability to pattern-match will help speed you up.

With the limitations posed by this question, we COULD break the numbers down into smaller groups and then total up all of these smaller numbers (it's a slightly longer way to do things, but if you don't immediately see the more complex calculations, you can still get to the correct answer with a bit of "hand math").

Let's start with making the first 2 digits the same...
66_
77_
88_
99_

Since the third digit has to be DIFFERENT from the matching pair, we have 9 options for each of the 4 groups above (you CAN'T count 666, 777, 888 or 999 - the numbers don't fit the restrictions).

Total of this group = 36

Next, let's make the first and third digits the same...
6_6
7_7
8_8
9_9

Here, we have a similar situation to the one we had above; we have 9 options for each of the 4 groups (you CAN'T count 666, 777, 888 or 999).

Total of this group = 36

Finally, let's make the second and third digits the same (I'll refer to those digits with the variable X)...
6XX
7XX
8XX
9XX

In this grouping, we have 1 "catch" - X can be any digit, BUT the number 600 is NOT permissible, since the prompt tells us for numbers GREATER THAN 600.

So, 6XX has 8 possibilities (you CAN'T count 600 or 666)
7XX, 8XX and 9XX have 9 possibilities each (you CAN'T count 777, 888 or 999)

Total of this group = 35

Overall total = 36 + 36 + 35 = 107

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

pacifist85 · Answer

Many very good approaches here! Good job everyone!

I did it in the amature's way:

So, I started calculating for 600, like this:
606
611
616
622 - 633 - 644 - 655 - 677 - 688 - 699
626 - 636 - 646 - 656 - 676 - 686 - 696

So, from here I only did 17*4= 68. By 4 because we are interested in 600, 700, 800 and 900.

Then I added the numbers after 660, that I had left out so that I wouldn't get confused:
660, 661, 662, 663, 664, 665, 667, 668, 669. This is 9*4= 36.

Adding 68+36= 104.

I saw that I was missing 3 from answer D. So, I realised that I didn't add the 600 (not allowed), 700, 800, 900.

So, 104+3 = 107.

It took a little more that 2 minutes, but again less than 2 1/2... You can't learn everything at the same time (unfortunately), so this solution kept me satisfied...

vihavivi · Answer

Hi Bunuel,

Can you please tell me what is wrong with my approach? Here is the way that i did:

The hundreds digit can be 6,7,8 or 9 => 4C1
Pick one more value for either tens or unit digit: 9C1
Choose 1 of 2 values above to be the repeated digits: 2C1
The permutation of the 3-digit integer: 3!/2!

==> 4C1*9C1*2C1*(3!/2!) = 4*9*2*3 = 216

216 -1 = 215 (1 is 600)

Thank you

EMPOWERgmatRichC · Answer

Hi vihavivi,

The big error in your calculation is in how you deal with the 'permutation'...

You described it in this way:

"The permutation of the 3-digit integer: 3!/2!"

However, this calculation includes options that are NOT allowed. For example, if you have the digits 4, 4 and 6, you could have 3 possible values: 446, 464 and 644. However, two of those values (446 and 464) are NOT greater than 600, so they should not be included.

GMAT assassins aren't born, they're made,
Rich

studentsensual · Answer

It works well for me, hope it helps u too:

600 700 800 900
 ab c 9 9 9 9
a bc  8 9 9 9 
 a b c  9 9 9 9

9*12=108-1=107

Arsh4MBA · Answer

I used bunuel's approach here.

Numbers between 600 & 999 =399.
Numbers with distinct digits =6*9*8 =288
Numbers with all 3 digits alike =4

So Numbers with 2 digits alike =399-(288+4) = 107

exc4libur · Answer

range: 601 to 999, inclusive

X={6,7,8,9}

XYY: 600 to 699 - 666 = 9 (#600,611,622...)
XYX: 606 to 696 - 666 = 9 (#606,616,626...)
XXY: 660 to 669 - 666 = 9 (#660,661,662...)

Total: 4(9*3)=108-1=107 (subtract #600)

Ans (D)

Hoozan · Answer

1) where the hundreds digit is a 6 the tens digit is a 6 and the units digit can be any other digit but a 6

1 * 1 * 9 = 9

Now we can have the non repeated digit in the tens place as well

9 * 2 =  18

2) where the hundreds digit is a 6 while the units digit is any digit but a 6 and the tens digit is same as the units digit

1 * 1 *  8

here we considered only 8 possible options for the units and tens place because we cannot have 600 as an option since the question has told us that we need a number greater than 600.

3) where the hundreds digit can be a 7/8/9 while the units digit is any other digit

3 * 1 * 9 = 27

Now we can have the non repeated digit in the tens place as well

27 * 2 =  54

4) where the hundreds digit can be a 7/8/9 while the units digit can be any other digit and the tens digit is same as the units digit

3 * 1 * 9 =  27

Adding all the possible outcomes:

18 + 8 + 27 + 54 =    107

Bunuel  's method is far more efficient where he basically does   -   =

Iq · Answer

the first number can only be on of 6,7,8,9

1. If the repeated number is amongst these, choices for filling each slot: 4 * 1 * 9 * 2 (since it can 6x6 or 66x) = 72
2. If the repeated number is not among 6,7,8,9, choices for each slot are: 4 * 9 * 1 = 36

But, 600 needs to be removed from the count, so: 72 + 36 - 1 =  107

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