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:

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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.