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the principle of coounting?

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Joined: 21 Feb 2010
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the principle of coounting? [#permalink] New post 04 Jul 2011, 09:01
hi all,
is there a quicker way to solve this problem? Thanks!

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

(C) 2008 GMAT Club - m01#10

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. Case 2: There are 10 ways to place the second digit, i.e. 0-9. Remember that we have counted 07 already. Thus, for two-digit numbers we have: 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.
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Re: the principle of coounting? [#permalink] New post 05 Jul 2011, 02:36
Expert's post
Another approach:
7 in unit's place:
In every 10 consecutive numbers, 7 will appear once. Since we have 1000 numbers, we have 100 sequences of 10 consecutive numbers (1-10, 11-20 etc) so 7 will appear in unit's place 100 times.
7 in ten's place:
In every 100 consecutive numbers, 7 appears in ten's place 10 times (from 70 to 79). We have 10 sequences of 100 consecutive numbers (1-100, 101-200 etc) so we get that 7 will appear in ten's place 10*10 = 100 times.
In every 1000 numbers, 7 will appear in hundred's place 100 times (from 700 to 799).
Total = 100 + 100 + 100 = 300

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Re: the principle of coounting? [#permalink] New post 06 Jul 2011, 05:14
Very clever solution from the Veritas instructor!
Re: the principle of coounting?   [#permalink] 06 Jul 2011, 05:14
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