Lesson: Listing and Counting

Comment on Listing and Counting

Can you pl explain this - "In

Can you pl explain this - "In writing all of the integers from 1 to 300, how many times is the digit 1 used?"
I am confused by the solution.
1) I don't understand the conclusion 'all digits occur the same number of times'
2) is there a different way altogether to solve it, for example if were asked to count the number of zeros, this approach wont work right? 1) If we focus on the UNITS digits as we count to 300, we get a units digit of 1 every 10 numbers. So, 1/10 of the 300 numbers will have 1 in the units digit place. This gives us 30 1's so far.

If we focus on the TENS digits as we count to 300, we get a 1 in the TENS location 10 times out of every 100 numbers (e.g., 210, 211, 212, 213..., 219). So, 1/10 of the 300 numbers will have 1 in the TENS digit place. This gives us another 30 1's.

Finally, the numbers from 100 to 199 have a 1 in the HUNDREDS position. So, there are another 100 1's here.

TOTAL = 30 + 30 + 100 = 160

2) We could modify this technique to make it work for counting 0's.

Thank you.

Thank you.

Can you explain the solution in the case we had to find number of zeros used from 1 to 300.
According your above method I got - 30 + 9+ 9 + 1 = 49.. You bet.

You bet.

NEW question: In writing all of the integers from 1 to 300, how many times is the digit 0 used?

0's IN THE UNITS POSITION
We get a 0 in the UNITS position once every 10 values: 10, 20, 30, 40, . . . 280, 290, and 300
In other words, 1/10 of the 300 numbers from 1 to 300 have a 0 in the UNITS position.
1/10 of 300 = 30, so we have 30 zeros so far

0's IN THE TENS POSITION
We get a 0 in the UNITS position in the following formats:
10-
20-
30-
NOTE: the "-" represents a units digit.

For the first case of 10-, the last (units) digit can be 0, 1, 2, 3, 4, ... 9. So, there are 10 zeros used (in the TENS position) in numbers in the 10- format.

Likewise, there are 10 zeros used (in the TENS position) in numbers in the 20- format.

Finally, there's only 1 zero used (in the TENS position) in the 30- format (300).

So, the TOTAL number of 0's used = 30 + 10 + 10 + 1 = 51 (just like you have!)

For the second GRE Prep

For the second GRE Prep question, about houses on elm street. When I count out all possibilities I see why the answer is 24 phone lines. However, I don't understand, conceptually, why this is not a combinations question with a denominator of 2-factorial.

I would have thought: 2 slots
First slot: 6 options
Second slot: 4 options
I don't want to double count AZ and ZA, so I divide my total of 24 by 2-factorial.

Is this not a combinations problem? Hi aseiden,

Hi aseiden,

When you're breaking the task into stages (aka slots), you need to be clear what you're actually doing during that stage. For example, what do you mean by "Second slot: 4 options"?

If you state what you're doing in each of those stages, you'll see that there is no duplication. So, we need not divide our answer by 2.

To see what I mean, check out my new post: http://greprepclub.com/forum/gre-math-challenge-117-on-elm-street-there-... 