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What is the minimum number of avoids for Dota 2 never ever have a match?

This can be translated into a graph theory problem belonging to the general heading of extremal graph theory. Namely, consider the graph $G$ on $N$ vertices given by the players where there is an edge ...
• 425k
1 vote

Is this correct solution to arranging consecutive flowers?

Between two red flowers, there are three spaces (including ends) where clumps of white flowers can be inserted, $\square R \square R \square$ as $\;\;8,\;\; 7-1,\;\; 6-2,\;\;or\;\; 6-1-1$ Favorable ...
• 42.7k

Counting problem involving sums

Alternative approach: Consider the following table: \begin{array}{| r | r | r | r |} \hline \text{Range of} ~d & \text{Range of} ~a & \text{Range of} ~(a-d) & \text{...
• 36.9k
Accepted

On $(0,1)$-strings and counting

If the total number of $1$ digits is $2^k,$ then we can group the ones digits as $1+1+2+4+\dots+2^{k-1}$ in $k+1$ places, with one zero after, and $n-2^k-1$ zeros out after $1,1,2,\dots,2^{k-1}.$ This ...
• 178k
Accepted

Is this correct solution to arranging consecutive flowers?

I would count arrangements with at least six consecutive white flowers by conditioning on the longest white block: Case 1: Longest White block is $8$ (denoted $W^8$). There are three ways this could ...
• 40.9k
1 vote
Accepted

Counting problem involving sums

With the edits, your attempt is nearly correct. As you noted the correct answer is half of yours. The reason why your answer is off by half is because you have double counted scenarios. You counted ...
• 80.1k
1 vote

Vanishing of Stirling number of second kind $S(n,k)$ for $k>n$

Substituting $j= k-i$ we have $$S(n, k) = \frac {1}{k!} \sum_{j=0}^{k}(-1)^{k-j}{\binom k j}j^{n}$$ and that is (apart from the factor $1/k!$) the value of the $k$-th iterated forward difference of ...
• 116k

Combinatorial Proof of the $\sum_{k=1}^{n} k^2 \binom{n}{k} = n(n + 1) 2^{n - 2}$

You have a set of $n$ people and want to pick a nonempty subset of this set and then pick a leader and treasurer from this group (the leader and the treasurer could be the same person). In how many ...
• 2,578

• 3,288
1 vote