25.3: Exercises
- Page ID
- 54486
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Find the value of the factorial or binomial coefficient.
- \(5!\)
- \(3!\)
- \(9!\)
- \(2!\)
- \(0!\)
- \(1!\)
- \(19!\)
- \(64!\)
- \(\dbinom{5}{2}\)
- \(\dbinom{9}{6}\)
- \(\dbinom{12}{1}\)
- \(\dbinom{12}{0}\)
- \(\dbinom{23}{22}\)
- \(\dbinom{19}{12}\)
- \(\dbinom{13}{11}\)
- \(\dbinom{16}{5}\)
- Answer
-
- \(120\)
- \(6\)
- \(362880\)
- \(2\)
- \(1\)
- \(1\)
- \(\approx 1.216 \cdot 10^{17}\)
- \(\approx 1.269 \cdot 10^{89}\)
- \(10\)
- \(84\)
- \(12\)
- \(1\)
- \(23\)
- \(50388\)
- \(78\)
- \(4368\)
Expand the expression via the binomial theorem.
- \((m+n)^{4}\)
- \((x+2)^5\)
- \((x-y)^6\)
- \((-p-q)^5\)
- Answer
-
- \(m^{4}+4 m^{3} n+6 m^{2} n^{2}+4 m n^{3}+n^{4}\)
- \(x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x+32\)
- \(x^{6}-6 x^{5} y+15 x^{4} y^{2}-20 x^{3} y^{3}+15 x^{2} y^{4}-6 x y^{5}+y^{6}\)
- \(-p^{5}-5 p^{4} q-10 p^{3} q^{2}-10 p^{2} q^{3}-5 p q^{4}-q^{5}\)
Expand the expression.
- \((x-2y)^3\)
- \((x-10)^4\)
- \((x^2y+y^2)^5\)
- \((2y^2-5x^4)^4\)
- \((x+\sqrt{x})^3\)
- \(\left(-2\dfrac{x^2}{y}-\dfrac{y^3}{x}\right)^5\)
- \((\sqrt{2}-2\sqrt{3})^3\)
- \((1-i)^3\)
- Answer
-
- \(x^{3}-6 x^{2} y+12 x y^{2}-8 y^{3}\)
- \(x^{4}-40 x^{3}+600 x^{2}-4000 x+10000\)
- \(x^{10} y^{5}+5 x^{8} y^{6}+10 x^{6} y^{7}+10 x^{4} y^{8}+5 x^{2} y^{9}+y^{10}\)
- \(16 y^{8}-160 x^{4} y^{6}+600 x^{8} y^{4}-1000 x^{12} y^{2}+625 x^{16}\)
- \(x^{3}+3 x^{\frac{5}{2}}+3 x^{2}+x^{\frac{3}{2}}\)
- \(-32 \dfrac{x^{10}}{y^{5}}-80 \dfrac{x^{7}}{y}-80 x^{4} y^{3}-40 x y^{7}-10 \dfrac{y^{11}}{x^{2}}-\dfrac{y^{15}}{x^{5}}\)
- \(38 \sqrt{2}-36 \sqrt{3}\)
- \(-2-2 i\)
Determine:
- the first \(3\) terms in the binomial expansion of \((xy-4x)^{5}\)
- the first \(2\) terms in the binomial expansion of \((2a^2+b^3)^{9}\)
- the last \(3\) terms in the binomial expansion of \((3y^2-x^2)^{7}\)
- the first \(3\) terms in the binomial expansion of \(\left(\dfrac{x}{y}-\dfrac{y}{x}\right)^{10}\)
- the last \(4\) terms in the binomial expansion of \(\left(m^3n+\dfrac{1}{2}n^2\right)^{6}\)
- Answer
-
- \(x^{5} y^{5}-20 x^{5} y^{4}+160 x^{5} y^{3}\)
- \(512 a^{18}+2304 a^{16} b^{3}\)
- \(-189 x^{10} y^{4}+21 x^{12} y^{2}-x^{14}\)
- \(\dfrac{x^{10}}{y^{10}}-10 \dfrac{x^{8}}{y^{8}}+45 \dfrac{x^{6}}{y^{6}}\)
- \(\dfrac{5}{2} m^{9} n^{9}+\dfrac{15}{16} m^{6} n^{10}+\dfrac{3}{16} m^{3} n^{11}+\dfrac{1}{64} n^{12}\)
Determine:
- the \(5\)th term in the binomial expansion of \((x+y)^{7}\)
- the \(3\)rd term in the binomial expansion of \((x^2-y)^{9}\)
- the \(10\)th term in the binomial expansion of \((2-w)^{11}\)
- the \(5\)th term in the binomial expansion of \((2x+xy)^{7}\)
- the \(7\)th term in the binomial expansion of \((2a+5b)^{6}\)
- the \(6\)th term in the binomial expansion of \((3p^2-q^3p)^{7}\)
- the \(10\)th term in the binomial expansion of \(\left(4+\dfrac{b}{2}\right)^{13}\)
- Answer
-
- \(35 x^{3} y^{4}\)
- \(36 x^{14} y^{2}\)
- \(-220 w^{9}\)
- \(280 x^{7} y^{4}\)
- \(15625 b^{6}\)
- \(-189 p^{9} q^{15}\)
- \(\dfrac{715}{2} b^{9}\)
Determine:
- the \(x^3y^{6}\)-term in the binomial expansion of \((x+y)^{9}\)
- the \(r^4s^4\)-term in the binomial expansion of \((r^2-s)^{6}\)
- the \(x^{4}\)-term in the binomial expansion of \((x-1)^{11}\)
- the \(x^3y^{6}\)-term in the binomial expansion of \((x^3+5y^2)^{4}\)
- the \(x^{7}\)-term in the binomial expansion of \((2x-x^2)^{5}\)
- the imaginary part of the number \((1+i)^3\)
- Answer
-
- \(84 x^{3} y^{6}\)
- \(15 r^{4} s^{4}\)
- \(-330 x^{4}\)
- \(500 x^{3} y^{6}\)
- \(80 x^{7}\)
- \(2 i\)