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# How many times will the digit 7 be written when listing the

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How many times will the digit 7 be written when listing the [#permalink]  21 Aug 2008, 12:07
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How many times will the digit 7 be written when listing the integers from 1 to 1000?

110
111
271
300
304

This problem is from GMACLUB test. I agree with answer but not the explanation.

Let see how others will tackle this problem.
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Last edited by x2suresh on 21 Aug 2008, 13:01, edited 1 time in total.
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]  21 Aug 2008, 12:16
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x2suresh wrote:
How many times will the digit 7 be written when listing the integers from 1 to 1000?

110
111
271
300
304

This problem is from GMACLUB test. I agree with answer but not the explanation.

Let see how others will tackle this problem.

D

I would do it by digits place.

Ones digit:
For every 100 numbers, 7 appears 10 times (x07, x17, x27, x37, x47, x57, x67, x77, x87, x97)
10*10 = 100

Tens digit:
For every 100 numbers, 7 appears 10 times (x70, x71, x72, x73, x74, x75, x76, x77, x78, x79)
10*10 = 100

Hundreds digit:
7 appears in the hundreds digit in every number from 700-799
799-700+1 = 100

Total 7s: 100+100+100 = 300
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]  21 Aug 2008, 12:18
I figured it like this: (and I realize there is a formula for this too, but I'm not sure what it is).

This method took me about 90 seconds to complete.

Units Digit: 10-7's per 100 numbers * 10 groups of 100 from 1 - 1000 = 100
Tens Digit: 10-7's in the 70's section of each group of 100 so 10*10 = 100
Hundreds digit: 100-7's from 700 to 799, so 100 again

Total = 300.

Am I forgetting anything?

x2suresh wrote:
How many times will the digit 7 be written when listing the integers from 1 to 1000?

110
111
271
300
304

This problem is from GMACLUB test. I agree with answer but not the explanation.

Let see how others will tackle this problem.

_________________

------------------------------------
J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a\$\$.

GMAT Club Premium Membership - big benefits and savings

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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]  21 Aug 2008, 12:42
I got D with similar counting method and took about 90 sec
But I would love to know some shorter method
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]  21 Aug 2008, 12:52
When counting the times a certain number appears, this might be a more scientific method of counting.

If you're counting units digits, you know that any single number will appear in the units digit column 1 time out of 10. If you're counting 1 to n, then multiply the number of times that digit appears out of 10 * n/10: Example: looking for # of 5s in the units column from 1 to 600.
There will be a 5 in each group of 10. So take 600 (the 1 to 600) and divide 600 by 10, for 60. There will be 60-5's in the units column from 1 to 600.

Tens column:
You see there will be 10-5's in the tens column per group of 100. Divide the last number by the size of the group. 600 / 100 = 6 groups, so 60-50s in the tens digit.

Hundreds column:
You know there will be 5's here only in 500-599. So that's 100 numbers per group of 1000. Since we don't go all the way to 1000, we know that's 1. so It's 100*1. what's the total?
60+60+100 = 220.

LOL....I'm not sure this makes it any easier!!

hibloom wrote:
I got D with similar counting method and took about 90 sec
But I would love to know some shorter method

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Re: GMAT CLUB - PS [#permalink]  21 Aug 2008, 13:09
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Expert's post
Short method. Consider slightly modified set: 000 - 999
We have 3000 digits. Frequency for 7 is 1/10. Therefore, answer is 300
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]  21 Aug 2008, 13:10
x2suresh wrote:
zoinnk wrote:
x2suresh wrote:
How many times will the digit 7 be written when listing the integers from 1 to 1000?

110
111
271
300
304

This problem is from GMACLUB test. I agree with answer but not the explanation.

Let see how others will tackle this problem.

For every 100 numbers, 7 appears 10 times (x07, x17, x27, x37, x47, x57, x67, x77, x87, x97) --> Do you belieive 7 appeared 10 times or 11 times here. Don't worry your answer is correct.. Here you treated x77 (7 in the 10th place ignored.. and reconsider when "Ten digit calculations" ) 10*10 = 100

Tens digit:
For every 100 numbers, 7 appears 10 times (x70, x71, x72, x73, x74, x75, x76, x77, x78, x79)
10*10 = 100

Hundreds digit:
7 appears in the hundreds digit in every number from 700-799
799-700+1 = 100

Total 7s: 100+100+100 = 300

I ignored the 7 in the tens place for that part of the calculation because i was just counting the 7s in the ones digit.
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Re: GMAT CLUB - PS [#permalink]  21 Aug 2008, 13:10
How would you do it counting zeros between 1 and 1000?

walker wrote:
Short method. Consider slightly modified set: 000 - 999
We have 3000 digits. Frequency for 7 is 1/10. Therefore, answer is 300

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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]  21 Aug 2008, 13:14
Hi Allen and Zoink,

Agree with both of you.. Now I agree with OE approach too..
OE is also explained similar to your approach. ( You ignored the x77 tenth digit 7 when counting unit digit calculations and reconsider this 7 when tenth digit calucations.. see zoink reply)

My approach was:

7 occur only once. (7XX, X7X,XX7)
= "7 is one of the digit" * "select other than 7" * "select other than 7" * (no of ways 7 can appear )
= 1*9*9* 3 =243
7 occur twice (77X,7X7,X77)
= "7 is two of the digit" * "select 3rd one other than 7" * ( Each number 7 written twice)

= 1*9*3 * 2 = 54
7 occur thrice = 1*3 (7 appeared 3 times)
=3

Sum = 243+54+3= 300.
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Last edited by x2suresh on 21 Aug 2008, 13:15, edited 1 time in total.
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Re: GMAT CLUB - PS [#permalink]  21 Aug 2008, 13:15
Expert's post
I considered, for example, 007 instead of 7 in order to have the same frequency for all digits
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Re: GMAT CLUB - PS [#permalink]  21 Aug 2008, 13:25
the way i did this is as follows..

we are really looking at 3 digit number..

X.Y.Z where x or y or z can be 7 or all of them could be 7..

howerver indepndently there are only 10 possibilities for x, 10 for y and 10 for z..

therefore there is only 1 possibility for x=7 and 1 possibility for y=7 and 1 possibility for z=7.

10.10.1 if z=7
10.1.10 if y=7
1.10.10 if x=7

total number of possibilities=100+100+100=300
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Re: GMAT CLUB - PS [#permalink]  21 Aug 2008, 14:32
Single digit nos.- 1 seven
Double digit nos.- 1*10 + 9*1 =19 sevens
3-digit nos.- 1*10*10 + 9*1*10 + 9*10*1 = 280 sevens
Total = 1 + 19 + 280 = 300 sevens
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Re: GMAT CLUB - PS [#permalink]  23 Aug 2008, 13:25
walker wrote:
Short method. Consider slightly modified set: 000 - 999
We have 3000 digits. Frequency for 7 is 1/10. Therefore, answer is 300

I think this is genius
Man you are really good
Re: GMAT CLUB - PS   [#permalink] 23 Aug 2008, 13:25
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