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## Digits, Numbers

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### Digits, Numbers

by vinay1983 » Mon Sep 30, 2013 6:59 am
I presume this will be easy to most of the "solving" people here

How many 4 digit positive integers do not contain the digit 3 or 6?

1. 2401
2. 3584
3. 4096
4. 5040
5. 7200
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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by Brent@GMATPrepNow » Mon Sep 30, 2013 7:15 am
vinay1983 wrote:I presume this will be easy to most of the "solving" people here

How many 4-digit positive integers do not contain the digit 3 or 6?

A. 2401
B. 3584
C. 4096
D. 5040
E. 7200
Take the task of building 4-digit positive integers and break it into stages.

Stage 1: Choose a thousands digit
This can be 1,2,4,5,7,8,or 9, so we can complete stage 1 in 7 ways

Stage 2: Choose a hundreds digit
This can be 0,1,2,4,5,7,8,or 9, so we can complete stage 2 in 8 ways

Stage 3: Choose a tens digit
This can be 0,1,2,4,5,7,8,or 9, so we can complete stage 3 in 8 ways

Stage 4: Choose a units digit
This can be 0,1,2,4,5,7,8,or 9, so we can complete stage 4 in 8 ways

By the Fundamental Counting Principle (FCP) we can complete all 4 stages (and thus build a 4-digit positive integer) in (7)(8)(8)(8) ways

IMPORTANT: we don't really need to calculate the product (7)(8)(8)(8)
We can just recognize that the units digit will be 4. That is (7)(8)(8)(8) = ---4
Since answer choice B, is the only one with units digit 4, it must be correct.

Cheers,
Brent

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by rakeshd347 » Mon Sep 30, 2013 8:11 am
vinay1983 wrote:I presume this will be easy to most of the "solving" people here

How many 4 digit positive integers do not contain the digit 3 or 6?

1. 2401
2. 3584
3. 4096
4. 5040
5. 7200
We have 4 places to fill to make 4 digit number ----
Now for the thousands digit we have only 7 options( we can't use 0,3 and 6).
For hundred tens and units digit we can use 8 digit( because we can't use 3 or 6 but we can use 0).
So the total number of 4 digit numbers without 3 or 6 is 7*8*8*8=3584

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by Zoser » Mon May 01, 2017 4:20 am
What would the answer be if the question said 3 AND 6 instead of 3 or 6?

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by Jay@ManhattanReview » Tue May 02, 2017 12:49 am
Zoser wrote:What would the answer be if the question said 3 AND 6 instead of 3 or 6?
Do you mean to ask, "How many 4 digit positive integers do not contain the digit 3 AND 6 together (not necessarily in order)?"

If yes, then let's follow this.

# of 4 digit positive integers do not contain the digit 3 AND 6 together = Total # of 4 digit positive integers - Total # of 4 digit positive integers that contain the digit 3 AND 6 together

Total # of 4 digit positive integers = 9*10*10*10 = 9(10^3) = 9000

"Total # of 4 digit positive integers that contain the digit 3 AND 6 together" can be computed the following way.

A. For '3' appears before '6':

a. Say '3' is in the thousands place and '6' in the hundreds place.

3 6 _ _ : 1*1*10*10 = 100

b. Say '3' is in the thousands place and '6' in the tens place.

3 _ 6 _ : 1*10*1*10 = 100

c. Say '3' is in the thousands place and '6' in the unit place.

3 _ _ 6 : 1*10*10*1 = 100

-----------------------

d. Say '3' is in the hundreds place and '6' in the tens place.

_3 6 _ : 9*1*1*10 = 90

e. Say '3' is in the hundreds place and '6' in the unit place.

_3 _ 6 : 9*1*10*1 = 90

------------------------

f. Say '3' is in the tens place and '6' in the unit place.

_ _3 6 : 9*10*1*1 = 90

-------------------------

Total # in which '3' appears before '6' = 100 + 100+ 100 + 90 + 90 + 90 = 570;

Similarly, total # in which '6' appears before '3' = 570;

Total # of 4 digit positive integers that contain the digit 3 AND 6 together = 570 + 570 = 1140

Thus, # of 4 digit positive integers do not contain the digit 3 AND 6 together = 9000 - 1140 = 7860.

Hope this is clear.

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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by ceilidh.erickson » Wed May 03, 2017 11:24 am
Zoser wrote:What would the answer be if the question said 3 AND 6 instead of 3 or 6?
Jay gave a good explanation, but... you're never actually going to see a question like this on the GMAT.

Questions that have vague or confusing wording are open to multiple interpretations, and as a result, they will yield bad data. Some high-scorers will get them wrong while low-scorers get them right, so it won't be a good indicator of ability level. Questions like this would be thrown out after experimentation (remember that a certain percentage of all questions that you'll see will be experimental). Since there's ambiguity about what "and" would mean in this context, it's not a question you'd see on the real test.

Broadly speaking, the combinatorics question that you're likely to see on the real GMAT are more straightforward and less tricky than a lot of the ones floating out there on the internet (including on this forum). Ceilidh Erickson
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by Scott@TargetTestPrep » Wed May 10, 2017 11:29 am
vinay1983 wrote:I presume this will be easy to most of the "solving" people here

How many 4 digit positive integers do not contain the digit 3 or 6?

1. 2401
2. 3584
3. 4096
4. 5040
5. 7200
We need to determine how many 4-digit positive integers do not contain a 3 or a 6.

For the first digit (the thousands digit), we have the options of 1, 2, 4, 5, 7, 8, and 9, so there are 7 options. For the next digit (the hundreds digit), we have 8 options, since we can include 0. For the next digit (the tens digit), we have another 8 options. For the last digit (the ones digit), we again have 8 options.Thus, the 4- digit number can be selected in 7 x 8 x 8 x 8 ways.

I also agree with Brent: since our answer choices have all different units digits, we simply need to calculate the units digit of the product above.

Since 7 x 8 has a units digit of 6, 8 x 8 has a units digit of 4, and 6 x 4 has a units digit of 4, the answer must be 3584.