# Test: Binomial Series - Ambitious

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Question 1:   Find the value of $\frac{16!}{13!}$
$240$
$3360$
$0$
$\frac{16}{13}$
Question 2:   Find the value of $\frac{4!}{0!}$
$24$
$0$
$1$
Does not exist
Question 3:   Knowing that combinational coefficient ${C}_{r}^{n}=\frac{n!}{\left(n-r\right)!r!}$, find ${C}_{1}^{5}$
$5$
$1$
$0$
$25$
Question 4:   Simplify: $\frac{\left(3n\right)!}{\left(3n+2\right)!}$
$\frac{3n}{3n+2}$
$\frac{1}{{\left(3n+3\right)}^{2}}$
$\frac{1}{\left(3n+3\right)\left(3n+2\right)}$
$\frac{1}{\left(3n+2\right)\left(3n+1\right)}$
Question 5:   Find the coefficient of ${x}^{4}$in the binomial expansion of ${\left(2-\frac{3}{x}\right)}^{9}$
$\frac{1}{{x}^{4}}$
$\frac{353682}{{x}^{4}}$
$\frac{647378}{{x}^{4}}$
$\frac{326592}{{x}^{4}}$
Question 6:   Find the $7$th term of ${\left(3-\frac{x}{4}\right)}^{13}$in the binomial expansion
$\frac{938223{x}^{6}}{1024}$
$\frac{2}{5}{x}^{6}$
$\frac{1}{3}{x}^{6}$
$\frac{3}{7}{x}^{6}$
Question 7:   Write down the sum of the first n terms using sigma notation of the following series: $-1+2-3+4-5+6-7+8...$
$\sum _{n=1}^{n}{\left(-1\right)}^{n}n$
$\sum _{n=1}^{n}\left(-1\right)n$
$\sum _{n=1}^{n}-n$
$\sum _{n=1}^{n}\left(-1\right){n}^{n}$
Question 8:   Find the value of $\sum _{n=1}^{70}n$
$140$
$2485$
$70$
$2500$
Question 9:   Find the value of $\sum _{r=1}^{n}\left(18r+3\right)$
$n\left(3{n}^{2}+3\right)$
$18{n}^{2}+4$
$3n\left(3{n}^{2}+4\right)$
$9{n}^{2}+3$
Question 10:   Knowing binomial expansion of ${e}^{x}$find ${e}^{0.41}$ by calculating first $5$ terms in the series
$1.57568$
$1.5067144$
$1.497722$
$1.603416$