How many four-digit numbers that do not contain the digits 3 or 6 are

Question

How many four-digit numbers that do not contain the digits 3 or 6 are there?

A. 2401
B. 3584
C. 4096
D. 5040
E. 7200

Actual solution states that the first digit has 7 possibilities which excludes 0,3,& 6. And the other 3 digits have 8 possibilities. So the total possibilities are 7 * 8 * 8 * 8==3584. This solution is perfect but why can't the other 3 digits have deceasing possibilities. The first digit has 7 possibilities. The second digit has 8 possibilities. So after selecting the second digit the third digit can have only 7 possibilities. similarly 4th digit could have 6 possibilities.

Why can't this be possible ? Could someone please clarify ?

Why can't this be possible ? Could someone please clarify ?

Bunuel · Answer

Because we are not told that the digits must be distinct hence they can repeat.

If it were: "how many 4-digit numbers are there which do not contain 3 or 6 and have all distinct digits?" then the answer would be - 7*7*6*5. The first digit can tale 7 values (except 0, 3, and 6), the second digit can take also 7 values (except 3, 6 and the one we used for the first digit), and so on.

Hope it's clear.

FacelessMan · Answer

total numbers = 7*8*8*8 = 3584. Hence B.

Abhishek009 · Answer

1st Digit can be filled up by the numbers - { 1 , 2 , 4 , 5 , 7 , 8 , 9 } = 7 ways

2nd Digit can be filled up by the numbers - { 0, 1 , 2 , 4 , 5 , 7 , 8 , 9 } = 8 ways

3rd Digit can be filled up by the numbers - { 0, 1 , 2 , 4 , 5 , 7 , 8 , 9 } = 8 ways

So, total No of ways is 7 * 8 * 8 => 3584
 
Hence answer will be (B)

Ashishsteag · Answer

Bunuel,
How will the answer be different if the question asks about 4 digit number that does not contain the digits 3 and 6??

I started this question wrong by taking cases such as:Only 3 but not 6 and Only 6 but not 3.

twister68 · Answer

Bunuel wrote:
How many four-digit numbers that do not contain the digits 3 or 6 are there?

A. 2401
B. 3584
C. 4096
D. 5040
E. 7200

Bunuel,
How will the answer be different if the question asks about 4 digit number that does not contain the digits 3 and 6??

I started this question wrong by taking cases such as:Only 3 but not 6 and Only 6 but not 3.

Exactly.. I also did it like this. There is clearly a difference b/w 3 or 6; 3 and 6.

BrentGMATPrepNow · Answer

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.

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

RELATED VIDEOS
 https://www.youtube.com/watch?v=w5d5bvCRyd4

https://www.youtube.com/watch?v=_PFBp2BoMUY

Rebaz · Answer

Can some please explain what the difference would have been had it been 3 AND 6 instead 3 OR 6?

When i saw 3 Or 6, this is how i interpreted and solved this question:

First taking care of the 3.

First digit: i had 8 possible digits (Excluding 0 and 3)

Second digit: i had 9 possible digits (Excluding just 3)

Third digit: i had 9 possible digits (Excluding just 3)

Fourth digit: i had 9 possible digits (Excluding just 3)

The above operations would result in: 8*9*9*9= 5832

Second taking care of the 6.

First digit: i had 8 possible digits (Excluding 0 and 6)

Second digit: i had 9 possible digits (Excluding just 6)

Third digit: i had 9 possible digits (Excluding just 6)

Fourth digit: i had 9 possible digits (Excluding just 6)

The above operations would result in: 8*9*9*9= 5832

The question was 3 or(which means +) 6.

Finally i added 5832+5832=11664.

But what i have noticed is that folks who solved this question have interpreted the question a though it was 3 AND 6 while we see 3 or 6.

I really appreciate if some one elucidates this for me!

Thanks in advanced!

bumpbot · Answer

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How many four-digit numbers that do not contain the digits 3 or 6 are

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