Found problems: 30
Prove the equality
$${n \choose 0}^2+ {n \choose 1}^2+ {n \choose 2}^2+...+{n \choose n}^2={2n \choose n}$$
Show that the sum of the first $n$ terms in the binomial expansion of $(2-1)^{-n}$ is $\frac{1}{2},$ where $n$ is a positive integer.
How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ?
(A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above.
Note: $C^{m}_{n}$ stands for $\binom {m}{n}$
Let $n\in\mathbb{N}$, the set $A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\}$ and the function $$f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}.$$
Prove that $f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).$
Determine all pairs of natural numbers $(m, r)$ with $2014 \ge m \ge r \ge 1$ that fulfill
$\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r} $