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How many times will the digit 7 be written when listing the [#permalink]
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21 Aug 2008, 13:07
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This topic is locked. If you want to discuss this question please repost it in the respective forum. 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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Re: GMAT CLUB  PS. (Dont agree with OE but agree with OA) [#permalink]
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21 Aug 2008, 13:16
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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 700799 799700+1 = 100 Total 7s: 100+100+100 = 300



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Re: GMAT CLUB  PS [#permalink]
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21 Aug 2008, 14:09
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]
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21 Aug 2008, 13:18
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. Answer: 300 Units Digit: 107's per 100 numbers * 10 groups of 100 from 1  1000 = 100 Tens Digit: 107's in the 70's section of each group of 100 so 10*10 = 100 Hundreds digit: 1007'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.
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Re: GMAT CLUB  PS. (Dont agree with OE but agree with OA) [#permalink]
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21 Aug 2008, 13:42
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]
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21 Aug 2008, 13:52
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 605's in the units column from 1 to 600. Tens column: You see there will be 105's in the tens column per group of 100. Divide the last number by the size of the group. 600 / 100 = 6 groups, so 6050s in the tens digit. Hundreds column: You know there will be 5's here only in 500599. 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. (Dont agree with OE but agree with OA) [#permalink]
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21 Aug 2008, 14:10
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 700799 799700+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]
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21 Aug 2008, 14:10
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]
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21 Aug 2008, 14:14
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, 14:15, edited 1 time in total.



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Re: GMAT CLUB  PS [#permalink]
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21 Aug 2008, 14:15
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]
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21 Aug 2008, 14:25
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]
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21 Aug 2008, 15:32
Single digit nos. 1 seven Double digit nos. 1*10 + 9*1 =19 sevens 3digit 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]
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23 Aug 2008, 14:25
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




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