Practice (83)
Let $n$ be a positive integer, show that $(3^{3n}-26n-1)$ is divisible by $676$.
Label the first row of the Pascal triangle as row $0$. How many odd numbers are there in the $2019^{th}$ row?
Find the coefficient of the $x$ term after having expanded $$(x^2+3x+2)^5$$
Find the constant term after $\left(\mid x\mid +\frac{1}{\mid x \mid} -2\right)^2$ is expanded.
Let $n$ be a positive integer. Show that $\left(3^{4n+2} + 5^{2n+1}\right)$ is divisible by $14$.
What is the remainder when $2021^{2020}$ is divided by $10^4$?
Show that $$1+4\binom{n}{1} + 7\binom{n}{2}+\cdots+(3n+1)\binom{n}{n}=(3n+2)\cdot 2^{n-1}$$
Simplify the expression $$\binom{2020}{0}^2 + \binom{2020}{1}^2 + \cdots + \binom{2020}{2020}^2$$
Show that $$\sum_{k=0}^m \binom{n}{k}\binom{n-k}{m-k}= 2^m\binom{n}{m}$$
Find the value of $(2 + \sqrt{5})^{1/3} - (-2 + \sqrt{5})^{1/3}$.
How many terms with odd coefficients are there in the expanded form of $$((x+1)(x+2)\cdots(x+2015))^{2016}$$
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?
Let $a_n=\binom{200}{n}(\sqrt[3]{6})^{200-n}\left(\frac{1}{\sqrt{2}}\right)^n$, where $n=1$, $2$, $\cdots$, $95$. Find the number of integer terms in $\{a_n\}$.
Compute the value of $$\sum_{n=2019}^\infty\frac{1}{\binom{n}{2019}}$$
Show that $$\binom{n}{1}-\frac{1}{2}\binom{n}{2}+\frac{1}{3}\binom{n}{3}-\cdots+(-1)^{n+1}\binom{n}{n}= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$
Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$
Show that $$\sum_{k=0}^n\left(2^k\binom{n}{k}\right)=3^n$$
Assuming a small packet of mm’s can contain anywhere from $20$ to $40$ mm’s in $6$ different colours. How many different mm packets are possible?
Show that $$\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1}=\binom{2n}{n-1}$$
Simplify $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} +\binom{n}{2} - \frac{1}{2}\binom{n}{3} + \cdots $$
Let $n > k$ be two positive integers. Simplify the following expression $$\binom{n}{k} + 2\binom{n-1}{k} + 3\binom{n-2}{k} + \cdots+ (n-k+1)\binom{k}{k}$$
Let positive integers $m$ and $n$ satisfy $m\le n$. Prove $$\sum_{k=m}^n\binom{n}{k}\binom{k}{m}=2^{n-m}\binom{n}{m}$$
Show that $$\sum_{k=0}^{2n-1}(-1)^k(k+1)\binom{2n}{k}^{-1}=\frac{1}{\binom{2n}{0}}-\frac{2}{\binom{2n}{1}}+\cdots-\frac{2n}{\binom{2n}{2n-1}}=0$$
Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
Compute $$\binom{2022}{1} - \binom{2022}{3} + \binom{2022}{5}-\cdots + \binom{2022}{2021}$$