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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 ?
Re: How many four-digit numbers that do not contain the digits 3 or 6 are
[#permalink]
02 Feb 2012, 14:06
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Expert Reply
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.
Re: How many four-digit numbers that do not contain the digits 3 or 6 are
[#permalink]
06 Dec 2018, 16:19
2
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Expert Reply
Top Contributor
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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