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I have four positive numbers $a_1,\dots,a_4$, each less than $45$. How many different ways are there for $a_1+a_2+a_3+a_4<90$? I require different permutations i.e $a_1a_2a_3a_4$ is different from $a_2a_1a_4a_3$

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    Any numbers? Perhaps you mean integers, or positive integers? Your question is ambiguous. If any numbers are allowed, there are infinitely many combinations.2012-11-09
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    Also, would you consider $a_1=21$, $a_2=a_3=a_4=20$ to be a different answer than $a_2=21,$ $a_1=a_3=a_4=20$, or the same thing? That makes a **crucial** difference. It looks like you probably interpret them to be different, given the `permutations` tag. Is this correct?2012-11-09
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    it should be positive integers2012-11-09
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    Rolled back: The previous edit introduced 3 changes: each a_i should be *positive*, each a_i is *less than* 45, the sum of the a_i's should be *less than* 90.2012-11-09
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    Related: [Four positive numbers less than 10 must add up to 12](http://math.stackexchange.com/questions/372624/probability-of-random-integers-digits-summing-to-12).2013-08-15

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