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I have the following problem:

Calculate how many numbers between $1$ and $1000$, inclusive, are even, or are between $100$ and $200$ or ends with $5$ or with $6$.

But I am confused with the wording of the problem as it seems there are many conditional conditions. Can anyone assist me in understanding the problem and how to solve it or is is there really a problem with the instructions?

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  • $\begingroup$ @fleablood Sorry for that, error in translation. I made the corrections. $\endgroup$ – Omari Celestine Dec 31 '17 at 21:14
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    $\begingroup$ This is a good time to use Inclusion-Exclusion. Let $E$ be the set of even numbers between $1$ and $1000$. Let $B$ be the set of numbers between $100$ and $200$. Let $D$ be the set of numbers between $1$ and $1000$ which end with a $5$ or with a $6$. You are asked to calculate $|E\cup B\cup D|$. $\endgroup$ – JMoravitz Dec 31 '17 at 21:19
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    $\begingroup$ To continue, inclusion-exclusion expands the above to be $|E|+|B|+|D|-|E\cap B|-|E\cap D|-|B\cap D|+|E\cap B\cap D|$, each of these terms should be much easier to manage, for example $|B\cap D|$ counts the number of numbers between $100$ and $200$ which ends in a $5$ or a $6$, i.e. $B\cap D=\{105,106,115,116,125,126,\dots,196\}$ which can be counted by hand or you can spot the pattern. $\endgroup$ – JMoravitz Dec 31 '17 at 21:23
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    $\begingroup$ use your rule of product. They will all be three digit integers which end in a $5$ or a $6$. Pick whether it ends in a five or a six. Pick what the hundred's digit is. Pick what the ten's digit is. $\endgroup$ – JMoravitz Dec 31 '17 at 21:26
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    $\begingroup$ Numbers either end in $0,1,2,3,4,5,6,7,8,9$. How many end in each? $\endgroup$ – fleablood Dec 31 '17 at 21:26
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Answer simpler questions first.

i) How many numbers are there between $1$ and $1000$ total?

ii) How many of them are odd? How many are even? How many are neither? How many are both?

iii) How many end with $6$? How many do not end in $6$.

iv) How many end with $5$? How many do not end in $5$?

v) How many end with either $5$ or with $6$.

vi) How many that end with $5$ are even? How many that end with $6$ are even? How many that end with $5$ are not even? How many that end with $6$ are not even?

vii) If you take all the numbers that are even and all the numbers that end with $5$ and all the numbers that end in $6$ how many numbers total would that be?

Can you answer those questions? If not where do you run into trouble?

P.S. I forgot to ask about between $100$ and $200$. But... same idea. Just think this out slowly. It's hard to answer as it's hard to know where your difficulty lies.

Others have mentioned Inclusion-Exclusion principal that will certainly help you in the future. But I think you need to simply figure this out slowly on your own.

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  • $\begingroup$ I think that I should be able to solve it now. Thx. Is just that I did not interpret it like this. $\endgroup$ – Omari Celestine Dec 31 '17 at 21:29

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