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# How many times 7 in 1000 - Gmatclub m01

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Manager
Joined: 22 Jun 2010
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How many times 7 in 1000 - Gmatclub m01 [#permalink]  12 Aug 2010, 12:31
How many times will the digit 7 be written when listing the integers from 1 to 1000?

(C) 2008 GMAT Club - m01#10

* 110
* 111
* 271
* 300
* 304

I get stuck at some point...

units = 1
2digit = 19-1=18

So 1-100 = 19

but now...
700-800 = 100 is clear... I have seen the official answer and explanation, but maybe someone has a more intuitive way...

Thanks!
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Re: How many times 7 in 1000 - Gmatclub m01 [#permalink]  12 Aug 2010, 13:16
3
KUDOS
Expert's post
AndreG wrote:
How many times will the digit 7 be written when listing the integers from 1 to 1000?

(C) 2008 GMAT Club - m01#10

* 110
* 111
* 271
* 300
* 304

I get stuck at some point...

units = 1
2digit = 19-1=18

So 1-100 = 19

but now...
700-800 = 100 is clear... I have seen the official answer and explanation, but maybe someone has a more intuitive way...

Thanks!

Many approaches are possible. For example:

Consider numbers from 0 to 999 written as follows:
1. 000
2. 001
3. 002
4. 003
...
...
...
1000. 999

We have 1000 numbers. We used 3 digits per number, hence used total of 3*1000=3000 digits. Now, why should ANY digit have preferences over another? We used each of 10 digits equal # of times, thus we used each digit (including 7) 3000/10=300 times.

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Re: How many times 7 in 1000 - Gmatclub m01 [#permalink]  12 Aug 2010, 18:05
Probability haunts!!! Awesome Bunuel - if you get a chance I had another question I had asked earlier on distribution in my previous posts.. would appreciate an answer!

Here is the mortal's way of doing it

007 - 1 time

7 X = 9 times
X 7 = 8 times
77 = 2 times

7 X X = 81 x 2 times
X 7 X = 72 x 2 times
X X 7 = 72 x 2 times

7 7 X = 9 x 2 times
7 X 7 = 9 x 2 times
X 7 7 = 8 x 2 times

777 = 3 times

= 300 times whew!

Does anyone know of a way between my dumb and bunuel's genius way here?
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Manager
Joined: 22 Jun 2010
Posts: 57
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Kudos [?]: 25 [0], given: 10

Re: How many times 7 in 1000 - Gmatclub m01 [#permalink]  13 Aug 2010, 02:14
Thanks Bunuel, you are the best
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Re: How many times 7 in 1000 - Gmatclub m01 [#permalink]  01 Oct 2010, 10:39
mainhoon wrote:
Probability haunts!!! Awesome Bunuel - if you get a chance I had another question I had asked earlier on distribution in my previous posts.. would appreciate an answer!

Here is the mortal's way of doing it

007 - 1 time

7 X = 9 times
X 7 = 8 times
77 = 2 times

7 X X = 81 x 2 times
X 7 X = 72 x 2 times
X X 7 = 72 x 2 times
7 7 X = 9 x 2 times
7 X 7 = 9 x 2 times
X 7 7 = 8 x 2 times

777 = 3 times

= 300 times whew!

Does anyone know of a way between my dumb and bunuel's genius way here?

I had the same idea too, even if it's not very smart it works (and pretty fast also)
but am afraid you made a little mistake, the multiplication by 2 is only for the numbers which have 2 sevens.
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Re: How many times 7 in 1000 - Gmatclub m01 [#permalink]  31 Oct 2010, 09:35
1
KUDOS
mainhoon wrote:
Probability haunts!!! Awesome Bunuel - if you get a chance I had another question I had asked earlier on distribution in my previous posts.. would appreciate an answer!

Here is the mortal's way of doing it

007 - 1 time

7 X = 9 times
X 7 = 8 times
77 = 2 times

7 X X = 81 x 2 times
X 7 X = 72 x 2 times
X X 7 = 72 x 2 times

7 7 X = 9 x 2 times
7 X 7 = 9 x 2 times
X 7 7 = 8 x 2 times

777 = 3 times

= 300 times whew!

Does anyone know of a way between my dumb and bunuel's genius way here?

Dear Mainhoon, can you explain how you are getting 7X = 9 times? I would think this is 10 times as you can have 70,71,72 etc? Can you also walk through how you are getting X7 8 times and why is 77 separate, we would just count them with the rest no? sorry I am puzzled by this problem so a further explanation will be helpful.

From the gmat club explanation:

*

There are several ways to count the number of times 7 appears between 7 and 997. One way is to consider the number of 7's in single, double, and triple digit numbers separately.

One-digit numbers: 7 is the only one-digit number.

Two-digit numbers: 7 could be the first digit or the second digit. Case 1: 7 is the first digit. There are 9 ways to place 7 as the first digit of a two-digit number why is this, please explain. Case 2: There are 10 ways to place the second digit, i.e. 0-9. Remember that we have counted 07 already it is still 10 digits with 7 right because without it would be 9. Thus, for two-digit numbers we have: $$10 + 9 = 19$$ numbers that contain a 7.

Three-digit numbers: Use the knowledge from the previous two scenarios: each hundred numbers will contain one 7 in numbers such as 107 or 507 and also 19 other sevens in numbers such as 271 or 237. Thus a total of 20 sevens per each hundred and 200 sevens for 1000. Since we have 700's within the range, that adds another 100 times that a seven will be written for a total of 300 times.
Re: How many times 7 in 1000 - Gmatclub m01   [#permalink] 31 Oct 2010, 09:35
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# How many times 7 in 1000 - Gmatclub m01

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