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How many six-digit numbers are there? How many of them contain the digit 5? Note that the first digit of n-digit number is nonzero. Does this question want to count all numbers between 100000 and 999999?
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george357 said: Your logic makes sense, but why do we have two answers? Your answer is that we have 900,000 numbers while Plato answer is 472392 numbers. Someone of you gotta be wrong! What do you think Dan, we give the Trophy to Plato or Jomo? Click to expand... I wouldn't know. I haven't been following this thread that closely. Perhaps you should try to work on it some more yourself and ask a question about different steps they took. -Dan
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Jomo said: I am just using the subtraction operation. We are experts in knowing how numbers there are if you start with the number 1 1, 2, 3, 4, ..., 50 are 50 numbers. 1,2,3, ..., 2027 are 2027 numbers. 1,2, ..., 1046 are 1046 numbers. Now if want to know, for example, how many numbers there are for 23, 24, 25, ..., 50 we do the following. We know that 1,2,3,4, ..., 22 ,23, 24, ...,50 has 50 numbers. But we only want to know how many numbers there are from 23 up to 50. That is, we do not want to include the numbers 1, 2, 3, ..., 22---which are 22 numbers. We just compute 50-22 = 28 and we know that 23, ..., 50 has 28 numbers Click to expand... Your logic makes sense, but why do we have two answers? Your answer is that we have 900,000 numbers while Plato answer is 472392 numbers. Someone of you gotta be wrong! What do you think Dan, we give the Trophy to Plato or Jomo?
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george357 said: The question did not mention anything about \(\displaystyle 1\), \(\displaystyle 2\), and \(\displaystyle 3\). What do you mean by I do not want to include them? Click to expand... I am just using the subtraction operation. We are experts in knowing how numbers there are if you start with the number 1 1, 2, 3, 4, ..., 50 are 50 numbers. 1,2,3, ..., 2027 are 2027 numbers. 1,2, ..., 1046 are 1046 numbers. Now if want to know, for example, how many numbers there are for 23, 24, 25, ..., 50 we do the following. We know that 1,2,3,4, ..., 22 ,23, 24, ...,50 has 50 numbers. But we only want to know how many numbers there are from 23 up to 50. That is, we do not want to include the numbers 1, 2, 3, ..., 22---which are 22 numbers. We just compute 50-22 = 28 and we know that 23, ..., 50 has 28 numbers
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Jomo said: 1, 2, 3, ..., 999999 is a list of 999999 numbers. 1, 2, 3, ..., 999999, 100000, 100001, ..., 999999 is a list of 999999 numbers. You do not want to include 1, 2, 3, ..., 99999. So 100000, 100001, ..., 999999 has 999999-99999 = 900,000 numbers Click to expand... The question did not mention anything about \(\displaystyle 1\), \(\displaystyle 2\), and \(\displaystyle 3\). What do you mean by I do not want to include them? Plato said: Please learn to use correct terms. 2 is between 1 & 3 but, 2 is not between 2 & 4. If we count all six digit numbers that have \(\displaystyle \bf\text{ none of their digits is a 5}\) Then subtract that number from the number of six digit numbers we have the number that have at least one 5 as a digit. Click to expand... Why did you mention \(\displaystyle 2\) is between \(\displaystyle 1\) and \(\displaystyle 3\)?
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george357 said: The way to find the numbers between \(\displaystyle 100000\) to \(\displaystyle 199999\), I kinda understand it, but I don't understand the second part. Click to expand... Please learn to use correct terms. 2 is between 1 & 3 but, 2 is not between 2 & 4. If we count all six digit numbers that have \(\displaystyle \bf\text{ none of their digits is a 5}\) Then subtract that number from the number of six digit numbers we have the number that have at least one 5 as a digit.
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1, 2, 3, ..., 999999 is a list of 999999 numbers. 1, 2, 3, ..., 999999, 100000, 100001, ..., 999999 is a list of 999999 numbers. You do not want to include 1, 2, 3, ..., 99999. So 100000, 100001, ..., 999999 has 999999-99999 = 900,000 numbers
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Plato said: There are \(\displaystyle 9\cdot 10^5=900000\) six digit integers from 100000 to 999999. (Please note the from-to construction betweenness does no work here!) There are \(\displaystyle 8\cdot 9^5=472392\) six digit numbers that do not contain a five. Therefore there are \(\displaystyle 9\cdot 10^5-8\cdot 9^5=427608 \) six digit numbers that contain at least one five. Click to expand... The way to find the numbers between \(\displaystyle 100000\) to \(\displaystyle 199999\), I kinda understand it, but I don't understand the second part.
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george357 said: How many six-digit numbers are there? Click to expand... There are \(\displaystyle 9\cdot 10^5=900000\) six digit integers from 100000 to 999999. (Please note the from-to construction betweenness does no work here!) george357 said: How many of them contain the digit 5? Click to expand... There are \(\displaystyle 8\cdot 9^5=472392\) six digit numbers that do not contain a five. Therefore there are \(\displaystyle 9\cdot 10^5-8\cdot 9^5=427608 \) six digit numbers that contain at least one five.
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The first part of the question does.
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