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 Aug 11 comment A Permutations/Combinations Question and Inquiry on Good Source for Studying The Concept Thanks. The proof is pretty nice. I have a question about it though. You proved $$\sum_{i=0}^{n} \binom{n}{i} = 2^{n}$$. Consider the case where I want to toss a coin 4 times. How many distinct outcomes exist? The intuition is that we have 4 slots and each slot has 2 outcomes and so the answer is $$2^{4}$$. How can I link this intuition with the summation intuition? In other words, how do I interpret this question to lead me to the answer $$\sum_{i=0}^{4} \binom{4}{i}$$ Aug 11 accepted A Permutations/Combinations Question and Inquiry on Good Source for Studying The Concept Aug 11 comment A Permutations/Combinations Question and Inquiry on Good Source for Studying The Concept I see. So the issue with my line of reasoning is as follows: Consider scenario 3, which I count as 3*7*6. Here, if we have two condiments "a" and "b", the number 3*7*6 includes selection of "a" first and "b" second and "b" first and "a" second, hence inflating the count. Aug 11 asked A Permutations/Combinations Question and Inquiry on Good Source for Studying The Concept May 11 awarded Notable Question Jan 26 accepted Measure Theory Book for My Background / Need Jan 26 comment Measure Theory Book for My Background / Need @jmbejara Does this book also develop measure theory? I have no background in it. If it does not, which measure theory book will complement "First Look at Rigorous Probability Theory". Jan 26 asked Measure Theory Book for My Background / Need Jul 2 awarded Curious Apr 10 awarded Talkative Mar 26 accepted Partitions of Unity-Integration on Manifolds Mar 26 asked Partitions of Unity-Integration on Manifolds Mar 5 comment When can we use Fubini's Theorem? Sorry, Reimann integral Mar 5 comment When can we use Fubini's Theorem? According to Munkres this might not be true. He says that we should consider the case where a function is integrable but not continuous. Then the function might be integrable over the region but the iterated integral may not because the function may behave badly along a line. Mar 5 asked When can we use Fubini's Theorem? Feb 18 awarded Benefactor Feb 12 comment Existence of Integral (for a function similar to Thomae's Function) But if I make the trivial partition, how would things work out. The width would be 1. so Reiman sum would equal 1/q (because there is only 1 Rectangle). I want it to equal 0 (i think 0 is the correct answer). Feb 12 comment Existence of Integral (for a function similar to Thomae's Function) Oh. But I thought that in part b) of the question, since the question asks us to compute $\displaystyle \int_{y\in I} f(x,y)$, i just want to fix $x$ and integrate over y. Feb 12 comment Existence of Integral (for a function similar to Thomae's Function) I guess I am having difficulty because there are infinitely many rationals in the intervals of a partition and so the function seems to be taking the value 1/q (a fixed number) at infinitely many points. I don't see a way around this. Feb 12 comment Existence of Integral (for a function similar to Thomae's Function) The Thomae's function argument works if we integrate over x