Tagged Questions

35 views

Combinatorial Inequality

For any integer $n>1$ prove that, $$\large 2^n < {2n \choose n} < \frac{2^n}{\prod^{i=n-1}_{i=0}(1-\frac{i}{n})}$$ Now proving that the first term is smaller than the third term is ...
84 views

To prove the inequality

$${n \choose 0}+{n \choose 3}+{n \choose 6}+\cdots+{n \choose 3k}\le \dfrac13(2^n+2)$$ Where $3k\le n$ It looks similar to the expansion of $2^n$ and since every three terms is missing 2 so the ...
63 views

175 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N}$ are all constants.I have tried my ...
264 views

79 views

326 views

Putnam PigeonHole

This is from page 12 of Putnam and Beyond. Problem: Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that ...
153 views

Combinatorial inequality $\binom{n}{j}\leqslant 2^n$

I was trying to prove (or to find a counterexample) of the following inequality: $$\binom{n}{j}\leqslant 2^n$$ As I coudn't find a proof/counterexample, I tested some numbers and could see it ...
116 views

Comparing two binomial coefficient sums

Let $j \in N, n\in N, n>1, q\geq 2$. I would like to show that $$\sum_{j=\frac{n}{\ln n}}^{\sqrt n/2}(2j-n)^q{n \choose j}<\sum_{j=\sqrt n/2}^{{\frac n2}}(2j-n)^q{n \choose j}$$ Any help would ...
204 views

A combinatorial inequality

How can I prove $$\log \binom nk \leq k \left(1 +\log\frac{n}{k}\right)$$ where $\binom\cdot\cdot$ stands for combination. I tried to use stirling approximation but I couldn't get the inequality.
374 views

Equidistant Points in the Plane problem by Paul Erdos

This problem was actually posed by Erdos in Geombinatorics (Oct. 1994) and goes as follows: Let $f(n)$ be the largest integer for which there are $n$ distinct points $x_1,x_2,\dots,x_n$ in the plane ...
129 views

how to prove an integer inequality

Given a positive integer $n$, is there a simple way to see that $$(n+3)^{n+2}(2n+5)^{n+2}(n+1)^{n+2}(2n+1)^n \ge (n+2)^{2n+4}(2n+3)^{2n+2}$$ Any hint is welcome.