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Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite I received a word problem that goes like this. A local kindergarten is thinking of making posters that show all the different ways of adding two or more integers from 1 to 9 to get a sum of 10. If there is enough space on each poster for up to 50 possible solutions, how many posters will the school need to make? (Note: sums that contain the same number but in a different order are considered to be different; for example, 1 + 9 and 9 + 1 are two different solutions.) What is the answer to this problem, and more importantly, how do I solve it? combinatorics combinations share | cite | improve this question edited Sep 10 at 23:54 N. F. Taussig 39.9k 9 32 53 asked Sep 10 at 21:43 Jolly 3 1 closed as off-topic by Theoretical Economist, Adrian Keister, user99914, Xander Henderson, Deepesh Meena Sep 11 at 3:26 This question appears to be off-topic. The users who voted to clos...

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $[a_n,b_n]_ninmathbbN$ be a sequence of closed bounded intervals in $mathbbR$ such that $(forall ninmathbbN)([a_n+1,b_n+1]subset [a_n,b_n])$ e $lim_ntoinfty(b_n-a_n)=0$. Let $Ssubset [a_0,b_0]$ a set such that $(forall ninmathbbN)(Scap [a_n,b_n]neq emptyset )$. We know from the Nested Intervals Theorem that exists $sigmainmathbbR$ such that $bigcap _n=1^infty [a_n,b_n]=sigma$. My question: the element $sigma$ belongs to the set $S$? That is, is it true that $sigmain S$? I have not been able to prove either that it is false or that it is true. I tried to prove that $sigmain S$ by contradiction: Assume that $sigmanotin S$, then $Scap left(bigcap _n=1^infty [a_n,b_n]right)=emptyset $. This implies that $(forall xin S)left(xnotin bigcap _n=1^infty [a_n,b_n]right)$. Given $xin S$, we have $xnotin bigcap _n=1^infty [a_n,b_n]Rightarrow (exists ninmathbbN)(xnotin [a_n,b_n])$. Therefore, $(forall xin S)left(xnotin bigc...