a) Write the total number of permutations of set of n objects arranged in a circle. [1] b) Write the general term in the expansion (a+x)^n . [1] c) Write the series of e(1 x)\ [x < 1] [1] d) Write…

a) The total number of permutations of $n$ distinct objects arranged in a circle is given by $(n-1)!$.

b) The general term in the expansion of $(a+x{)}^n$ is given by: $$T_{r+1}=\left(\genfrac{}{}{0ex}{}{n}{r}\right)a^{n-r}{x}^r$$ where $r=0,1,2,\dots ,n$.

c) The series expansion of ${\log }_e(1-x)$ for $|x|<1$ is: $${\log }_e(1-x)=-\sum _{k=1}^{\infty }\frac{x^k}{k}=-\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\dots \right)$$

d) The complex number $(\cos \theta +i\sin \theta )$ in Euler's form is: $$e^{i\theta }$$

e) The sum of the cubes of the first $n$ natural numbers is: $$1^3+2^3+3^3+\cdots +n^3={\left(\frac{n(n+1)}{2}\right)}^2$$