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 21h answered Prove by induction: A tree on n≥2 vertices has ≥2 leaves 1d comment Prove by induction: A tree on n≥2 vertices has ≥2 leaves This proof is fine if you already know that every tree on $k+1$ vertices can be obtained by adding a vertex to some tree on $k$ vertices. Just how do you know that? 1d comment Combinations choosing equal groups results in division by 2 Do you think $6$ is the right answer to the last problem in my common? If you have $3$ people, call them $A,B,C,$ there are $6$ ways to choose two teams of $1$? OK, let's list them.1. A&B 2. A&C 3. B&A. 4. B&C. 5. C&A. 6. C&B. The reason I divided by two is that I didn't think A&B and B&A should be counted as two different ways of making two teams of one. Of course, if they are to be counted as different ways, then I don't divide by $2$ and the answer if $6.$ 1d comment Combinations choosing equal groups results in division by 2 Sometimes looking at a smaller example makes it easier to see what is going on. $3$ people, split into team of $2$ and team of $1,$ the number of ways is $\binom32\binom11=3\cdot1=3.$ Split them into two teams of $1,$ the number of ways is $\binom31\binom21/2=3\cdot2/2=3.$ Why? 1d revised Partition of natural number not equal to factorial added 30 characters in body 1d answered Partition of natural number not equal to factorial 2d comment Existence of how many sets is asserted by the axiom of choice in this case? What do you mean, "letting them all exist has its own problems"? It seems to be a necessary truth that they all exist, axiom of choice or no axiom of choice. 2d revised binomial coefficient where k > n added 12 characters in body 2d answered binomial coefficient where k > n Apr 28 comment What book about algebraic combinatorics is it? Why are you asking us instead of Dr. Balogh? Apr 28 comment Infinitely countable subset of $\mathbb{R}^2$ is connected. Or just draw a continuum of lines, each intersecting the segment aa' and perpendicular to it. Pick one of those parallel lines which misses the set $A.$ Apr 27 revised How many distinct patterns exist for a 5x5 grid by filling 3 colors? added 4 characters in body; edited title Apr 27 comment Minimum number of marked squares on $n × n$ board You only have to mark every 3rd row, right? So for a 6 by 6 board, you only have to make the squares in rows 2 and 5. If n is divisible by 3, then you can get by with marking $n^2/3$ instead of $n^2/2$ squares. Apr 27 comment Simple connected graph question "Simple" is irrelevant. If the graph has the minimum number of edges for connectivity, it won't have two vertices joined by two edges. Apr 27 comment How can we show that the adjacency matrix of a regular graph commutes with its complement What is the complement of a matrix?? Apr 26 revised How do I write $y''+y' +\sin y \cos y = 0$ as a first order system? added 32 characters in body Apr 26 answered How do I write $y''+y' +\sin y \cos y = 0$ as a first order system? Apr 25 answered Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded? Apr 25 revised Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded? added 14 characters in body Apr 25 comment Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded? If $d(x,y)$ is a metric, doesn't the metric $d_1(x,y)=\min\{1,d(x,y)\}$ induce the same topology as $d(x,y)$?