Show that $$\frac{1}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n-1+k}{n-1}x^k$$
Let $n$ be an odd positive integer, and $$N=6^n + \binom{n}{1}\cdot 6^{n-1}+\cdots + \binom{n}{n-1}\cdot 6-1$$
Find the remainder when $N$ is being divided by $8$.
Assuming that $$(1-2x)^7=a_0 + a_1x+a_2x^2+\cdots+a_7x^7$$
Find the value of
Let $m$ and $n$ be positive integers satisfying $1 < m < n$. Show that $(1+m)^n > (1+n)^m$.
Show that the following relation holds for any positive integers $1 < k \le m < n$: $$\binom{n}{k}m^k > \binom{m}{k}n^k$$
Let $\{a_n\}$ be a geometric sequence whose initial term is $a_1$ and common ratio is $q$. Show that $$a_1\binom{n}{0}-a_2\binom{n}{1}+a_3\binom{n}{2}-a_4\binom{n}{3}+\cdots+(-1)^na_{n+1}\binom{n}{n}=a_1(1-q)^n$$
where $n$ is a positive integer.
Find the constant term in the expansion of $\left(\frac{x}{2}+\frac{1}{x}+\sqrt{2}\right)^5$.
Let $n$ be a positive integer and the coefficient of the $x^3$ term in the expansion of $(1+\frac{x}{n})^n$ be $\frac{1}{16}$. Find $n$.
Let $p$, $q$, and $n$ be three positive integers, show that $$\sum_{k=0}^n\binom{p+k}{p}\binom{q+n-k}{q} = \binom{p+q+n+1}{p+q+1}$$
Calculate the value of $$\displaystyle\sum_{k=0}^{n}\frac{1}{2^k}\binom{n}{k}$$
Compute the value of $$\sum_{k=0}^{n}(-1)^k\frac{1}{k+1}\binom{n}{k}=\binom{n}{0}-\frac{1}{2}\binom{n}{1}+\frac{1}{3}\binom{n}{2} -\cdots+ (-1)^n\frac{1}{n+1}\binom{n}{n}$$
Show that $$\displaystyle\sum_{k=0}^{n}\binom{2n}{k} = 2^{2n-1}+\frac{1}{2}\binom{2n}{n}$$
Show that for any positive integer $n$, the value of $\displaystyle\sum_{k=0}^{n}2^{3k}\binom{2n+1}{2k+1}$ is not a multiple of $5$.
Let $\lfloor{x}\rfloor$ be the largest integer not exceeding real number $x$. Show that $$\sum_{k=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\left(\left(1-\frac{2k}{n}\right)\binom{n}{k}\right)^2=\frac{1}{n}\binom{2n-2}{n-1}$$
Let $n$, $r$, and $m$ all be positive integers, $r\le m$, and $\omega_k=e^{\frac{2k\pi}{m}i}$ be a complex root to the equation $x^m=1$. Show $$\sum_{k=0}^{\lfloor{\frac{n-r}{m}}\rfloor}\binom{n}{r+km}x^{r+km}=\frac{1}{m}\sum_{k=0}^{m-1}\omega^{-r}(1+x\omega_k)^n$$
where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding real number $x$.
Let $n$ be a positive integer and $N=\displaystyle\sum_{k=0}^{n}(-1)^k\binom{n}{k}^2$. Show that $N=0$ if $n$ is odd, and $N=(-1)^{\frac{n}{2}}\displaystyle\binom{n}{\frac{n}{2}}$ if $n$ is even.
Let $n$ be a positive integer and function $\lfloor{x}\rfloor$ return the largest integer not exceeding $x$. Compute the value of $$\sum_{k=0}^{\lfloor{\frac{n}{2}}\rfloor}\binom{n-k}{k}$$
Let $m$ and $n$ be two positive integers satisfying $m\le n$. Find the value of $$S_{m,n} = \displaystyle\sum_{k=0}^{m}(-1)^k\binom{m}{n}$$
Let $m$ and $n$ be two positive integers satisfying $m < n$. Show that $$S_{m,n}=\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0$$
Show that $$\sum_{k=0}^{n}(-1)^k\frac{m}{m+k}\binom{n}{k}=\frac{1}{\binom{m+n}{n}}$$
Show that $$\sum_{k=0}^{n}(-1)^k2^{2n-2k}\binom{2n-k+1}{k}=n+1$$
Let the binary representation of positive integer $n$ be $b_tb_{t-1}\cdots b_1b_0$. Show that $$\binom{n}{2^j} \equiv b_j \pmod{2}$$
where $j$ is a non-negative integer. Note that $\binom{n}{m} = 0$ if $m > n$.
Let $n$ be a positive integer and $k$ be the number of $1$s in $n$'s binary representation. Show there are $2^k$ odd integers in $\binom{n}{0}$, $\binom{n}{1}$, $\cdots$, $\binom{n}{n}$.
Show that $$\frac{2n+2}{2n+1}\cdot\frac{1}{\binom{2n}{k}}=\frac{1}{\binom{2n+1}{k}}+\frac{1}{\binom{2n+1}{k+1}}$$
Calculate the value of $$\sum_{k=1}^{2n-1}(-1)^{k-1}\binom{2n}{k}^{-1}$$