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Senior Manager  Joined: 23 May 2008
Posts: 298
Of the three digits greater than 700, how many have two digi  [#permalink]

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6
54 00:00

Difficulty:   85% (hard)

Question Stats: 55% (02:37) correct 45% (02:36) wrong based on 837 sessions

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Of the three digits 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

Is there a faster method to find out the answer, because i listed down all the numbers and it took me hardly 1.5 mins.
but what if there are more numbers ?

OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/of-the-three- ... 35188.html
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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19
18
This can be solved in another way and maybe it's faster:

Numbers between 700 and 999 =299;

Numbers with all distinct digits = 3*9*8=216;

Numbers with all the same digits more than 700 =3, (777, 888, 999);

Total: 299-216-3=80.
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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3
16
gurpreet07 wrote:
Of the three digits 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

Is there a faster method to find out the answer, because i listed down all the numbers and it took me hardly 1.5 mins.
but what if there are more numbers ?

Step by step:

The number can have three forms:

XXY meaning that the first digit is the repeated digit;
XYX meaning that the first digit is repeated digit;
and
YXX meaning that the first digit is not repeated digit.

XXY --> the repeated digits can be 7,8 or 9, so 3 possibilities and the third one can take 9 possible values total 3*9=27.

The same with XYX =3*9=27;

YXX --> the non repeated first digit can be 7, 8, 9, so 3 possibilities and the repeated digits can take 9 possible values, total 3*9=27. But this will give us the number 700 as well and we know that the number should be more than 700, hence 27-1=26.

TOTAL=27+27+26=80

Hope it helps.
Senior Manager  Joined: 23 May 2008
Posts: 298
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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1
1
Bunuel wrote:
gurpreet07 wrote:
Of the three digits 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

Is there a faster method to find out the answer, because i listed down all the numbers and it took me hardly 1.5 mins.
but what if there are more numbers ?

Step by step:

The number can have three forms:

XXY meaning that the first digit is the repeated digit;
XYX meaning that the first digit is repeated digit;
and
YXX meaning that the first digit is not repeated digit.

XXY --> the repeated digits can be 7,8 or 9, so 3 possibilities and the third one can take 9 possible values total 3*9=27.

The same with XYX =3*9=27;

YXX --> the non repeated first digit can be 7, 8, 9, so 3 possibilities and the repeated digits can take 9 possible values, total 3*9=27. But this will give us the number 700 as well and we know that the number should be more than 700, hence 27-1=26.

TOTAL=27+27+26=80

Hope it helps.

thnku....very fast and a efficient method.
Senior Manager  Joined: 23 May 2008
Posts: 298
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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1
Bunuel wrote:
This can be solved in another way and maybe it's faster:

Numbers between 700 and 999 =299;

Numbers with all distinct digits = 3*9*8=216;

Numbers with all the same digits more than 700 =3, (777, 888, 999);

Total: 299-216-3=80.

Hey Bunuel
thanks man...from where do you get such idea's
just awesome
Intern  Joined: 17 Dec 2009
Posts: 19
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Bunuel,

I don't understand how there is only 27 numbers based on your calculation. For example, XXY the first number has to be 7, 8 or 9 so 3 possibilities and the next number would also have to be 7, 8 or 9 but the last number could be 9 values but how is that equal to 3*9? Wouldn't that be 3*3*9? I'm don't understand.
Manager  Joined: 05 Mar 2010
Posts: 147
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Bunuel,

I don't understand how there is only 27 numbers based on your calculation. For example, XXY the first number has to be 7, 8 or 9 so 3 possibilities and the next number would also have to be 7, 8 or 9 but the last number could be 9 values but how is that equal to 3*9? Wouldn't that be 3*3*9? I'm don't understand.

in case of XXY, first digit is 7,8 or 9 then so 3 ways
2nd digit has be same as 1st so only one way
you can also write it as 3*1*9 = 27

Hope it helps
_________________
Success is my Destiny
Intern  Joined: 29 Mar 2010
Posts: 28
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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How have you guys become so confident, predicting :

700-999

XXY setup = t occurrences without a single calculation line etc...

Isnt there any mathematical way of solving it.
Manager  Joined: 07 Jan 2010
Posts: 134
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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wow Bunuel ...
I tried to formulate a pattern and reach the answer. But it seems I missed something ...well .... Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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3
shrivastavarohit wrote:
How have you guys become so confident, predicting :

700-999

XXY setup = t occurrences without a single calculation line etc...

Isnt there any mathematical way of solving it.

This is how I solved it:

Three digit number can have only following 3 patterns:
A. all digits are distinct;
B. two are alike and third is different;
C. all three digits are alike.

We need to calculate B. B=Total - A - C

Total numbers from 700 to 999 = 299.
A. all digits are distinct = 3(first digit can have only three values 7,8, or 9)*9*8=216;
C. all three are alike = 3 (777, 888, 999)

So, 299-216-3=80.

Another way to solve this problem is in my first post.

Hope it's clear.
Intern  Joined: 08 Jun 2010
Posts: 9
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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I am getting 78 as the answer. So I went with closest answer 80 (C).

This is how I arrived @ 78:

For numbers greater than 700 we have following possibilities:

a=> 7xx .. Here x being any digit 0-9, excluding 7 and 0 {0 is excluded because number should be >700}
b=> 7x7 .. Here x can be any digit from 0-9, excluding 7. {no 3 digits should be same}
c=> 77x .. Here x can be any digit from 0-9, excluding 7 for same reason as above.

a + b + c = 8 + 9 + 9 = 26

Hence same is true with 8xx , 8x8, 88x and 9xx,9x9, 99x ...

So summing up I got 78 ...

Please help me if I am wrong here. I am unable to figure out where did I miss 2 more numbers. Any pointers ?
Intern  Joined: 07 Jun 2010
Posts: 46
Location: United States
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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for 700 ==> 711 , 722 , ...... , 799 (except 777) = 8
,707, 717 , 727 , .... , 797 = 9
770, 771 , 772 , ........ , 779 = 9
total = 26

similarly for 800 and 900 series you will have 26 each

so 26 * 3 = 78

and 800 and 900 so 78 + 2 = 80
Manager  Joined: 21 Feb 2010
Posts: 167
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Bunuel wrote:
gurpreet07 wrote:
Of the three digits 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

Is there a faster method to find out the answer, because i listed down all the numbers and it took me hardly 1.5 mins.
but what if there are more numbers ?

Step by step:

The number can have three forms:

XXY meaning that the first digit is the repeated digit;
XYX meaning that the first digit is repeated digit;
and
YXX meaning that the first digit is not repeated digit.

XXY --> the repeated digits can be 7,8 or 9, so 3 possibilities and the third one can take 9 possible values total 3*9=27.

The same with XYX =3*9=27;

YXX --> the non repeated first digit can be 7, 8, 9, so 3 possibilities and the repeated digits can take 9 possible values, total 3*9=27. But this will give us the number 700 as well and we know that the number should be more than 700, hence 27-1=26.

TOTAL=27+27+26=80

Hope it helps.

thanks! this is an efficient way to solve it. is this method feasible for this kind of questions--fundamental counting?
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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1
Bunuel wrote:
Hope it helps.
tt11234 wrote:
thanks! this is an efficient way to solve it. is this method feasible for this kind of questions--fundamental counting?

I don't really understand your question...

Also there is another, easier way to solve this question, shown in my second post:

3-digit numbers more than 700 = 299;

Numbers with all distinct digits = 3*9*8=216 (first digit can take 3 values - 7, 8, 9; second digit can take 9 values and the third digit 8 values);

Numbers with all the same digits more than 700 = 3, (777, 888, 999);

{all} - {all distinct} - {all same} = {two equal} --> 299-216-3=80.

Manager  Joined: 09 Jun 2010
Posts: 50
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Thanks Bunuel for the effort
but I dont really understand why you choose 299, and what is the meaning of distinct, and why is there 3, 9 , 8?
Pleas explaing that to me
thanks again
Kaplan GMAT Instructor Joined: 21 Jun 2010
Posts: 66
Location: Toronto
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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xmagedo wrote:
Thanks Bunuel for the effort
but I dont really understand why you choose 299, and what is the meaning of distinct, and why is there 3, 9 , 8?
Pleas explaing that to me
thanks again

The question asks about three-digit integers greater than 700. So, all the numbers in the range 701 to 999, inclusive. There will be 999 - 701 + 1 = 299 such numbers in this range. (Whenever you want to count the number of objects in a range, subtract the smallest number from the greatest and add 1. For example, the number of integers in the range of 1 to 6 is 6 -1 +1 = 6).

"distinct" just means different. So, for example, in the number "799", the final two digits are NOT distinct. Here: 798, they are.
Intern  Joined: 07 Oct 2011
Posts: 25
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Numbers between 700 and 999 =299;

Numbers with all distinct digits = 3*9*8=216;

Numbers with all the same digits more than 700 =3, (777, 888, 999);

Total: 299-216-3=80.

I've a doubt. Numbers with all distinct digits = 3*9*8=216;

I feel 9 above is not right because if we take one of the 3 digits (7,8 or 9), then we shouldn't take repeated values (how is 9 possible).
Intern  Joined: 07 Oct 2011
Posts: 25
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Ok, I got it I forgot to take from 0-9 .. 10 digits

10-1=9
Manager  Joined: 20 Dec 2013
Posts: 222
Location: India
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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We can fix 2 digits in 3 ways ie 7,8,9 and the third digit can be fixed in 9 ways(10 digits-1 repeating digit)
Total=3*9=27
Since this is possible in 3 different ways(first digit diff from other two,second digit diff from other two and third digit diff from other two) we get 3* 27=81 but we subtract 1 for 700
Ans 80

Posted from my mobile device
Intern  Joined: 01 Feb 2016
Posts: 8
Re: Of the three digits greater than 700, how many have two digi  [#permalink]

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Repeating the explanation from above, the sequence of nos. should be XXY, XYX, YXX. As we need 3 digit nos. greater that 700, the 100th place in can be either of 7, 8 or 9 (3 ways to select 100th place digit)

Once 100th place if fixed, say units place is same as hundred place (XYX). this means that the tens place can be picked in 9 ways (all digits from 0-9 except the place in hundred place) and unit place can be picked in 1 way only (same as hundred place). So total ways for XYX is 3*9*1=27

We need to repeat this for XXY format. This would again generate 3*1*9 ways = 27

We need to repeat this now with YXX which will give 3.9.1 ways = 27. But we need to exclude 700 from this as we need nos greater than 700. This this combination will give 27-1=26 ways

Total ways = 27+27+26=80 Re: Of the three digits greater than 700, how many have two digi   [#permalink] 16 Jun 2016, 11:55

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