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Manager  Joined: 17 Sep 2011
Posts: 125
How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 69% (01:21) correct 31% (02:04) wrong based on 150 sessions

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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 ?

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Math Expert V
Joined: 02 Sep 2009
Posts: 58449
Re: How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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abhi47 wrote:
How many 4 digit nos that do not contain the digits 3 or 6 are there ?

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 ?

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.
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Re: How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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total numbers = 7*8*8*8 = 3584. Hence B.
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Re: How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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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

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)

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Re: How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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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.
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GMAT 1: 710 Q49 V38 Re: How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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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.
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Posts: 4009
Re: How many four-digit numbers that do not contain the digits 3 or 6 are  [#permalink]

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abhi47 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

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.

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_________________ Re: How many four-digit numbers that do not contain the digits 3 or 6 are   [#permalink] 06 Dec 2018, 17:19
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