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25.3: Exercises

  • Page ID
    54486
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    Exercise \(\PageIndex{1}\)

    Find the value of the factorial or binomial coefficient.

    1. \(5!\)
    2. \(3!\)
    3. \(9!\)
    4. \(2!\)
    5. \(0!\)
    6. \(1!\)
    7. \(19!\)
    8. \(64!\)
    9. \(\dbinom{5}{2}\)
    10. \(\dbinom{9}{6}\)
    11. \(\dbinom{12}{1}\)
    12. \(\dbinom{12}{0}\)
    13. \(\dbinom{23}{22}\)
    14. \(\dbinom{19}{12}\)
    15. \(\dbinom{13}{11}\)
    16. \(\dbinom{16}{5}\)
    Answer
    1. \(120\)
    2. \(6\)
    3. \(362880\)
    4. \(2\)
    5. \(1\)
    6. \(1\)
    7. \(\approx 1.216 \cdot 10^{17}\)
    8. \(\approx 1.269 \cdot 10^{89}\)
    9. \(10\)
    10. \(84\)
    11. \(12\)
    12. \(1\)
    13. \(23\)
    14. \(50388\)
    15. \(78\)
    16. \(4368\)

    Exercise \(\PageIndex{2}\)

    Expand the expression via the binomial theorem.

    1. \((m+n)^{4}\)
    2. \((x+2)^5\)
    3. \((x-y)^6\)
    4. \((-p-q)^5\)
    Answer
    1. \(m^{4}+4 m^{3} n+6 m^{2} n^{2}+4 m n^{3}+n^{4}\)
    2. \(x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x+32\)
    3. \(x^{6}-6 x^{5} y+15 x^{4} y^{2}-20 x^{3} y^{3}+15 x^{2} y^{4}-6 x y^{5}+y^{6}\)
    4. \(-p^{5}-5 p^{4} q-10 p^{3} q^{2}-10 p^{2} q^{3}-5 p q^{4}-q^{5}\)

    Exercise \(\PageIndex{3}\)

    Expand the expression.

    1. \((x-2y)^3\)
    2. \((x-10)^4\)
    3. \((x^2y+y^2)^5\)
    4. \((2y^2-5x^4)^4\)
    5. \((x+\sqrt{x})^3\)
    6. \(\left(-2\dfrac{x^2}{y}-\dfrac{y^3}{x}\right)^5\)
    7. \((\sqrt{2}-2\sqrt{3})^3\)
    8. \((1-i)^3\)
    Answer
    1. \(x^{3}-6 x^{2} y+12 x y^{2}-8 y^{3}\)
    2. \(x^{4}-40 x^{3}+600 x^{2}-4000 x+10000\)
    3. \(x^{10} y^{5}+5 x^{8} y^{6}+10 x^{6} y^{7}+10 x^{4} y^{8}+5 x^{2} y^{9}+y^{10}\)
    4. \(16 y^{8}-160 x^{4} y^{6}+600 x^{8} y^{4}-1000 x^{12} y^{2}+625 x^{16}\)
    5. \(x^{3}+3 x^{\frac{5}{2}}+3 x^{2}+x^{\frac{3}{2}}\)
    6. \(-32 \dfrac{x^{10}}{y^{5}}-80 \dfrac{x^{7}}{y}-80 x^{4} y^{3}-40 x y^{7}-10 \dfrac{y^{11}}{x^{2}}-\dfrac{y^{15}}{x^{5}}\)
    7. \(38 \sqrt{2}-36 \sqrt{3}\)
    8. \(-2-2 i\)

    Exercise \(\PageIndex{4}\)

    Determine:

    1. the first \(3\) terms in the binomial expansion of \((xy-4x)^{5}\)
    2. the first \(2\) terms in the binomial expansion of \((2a^2+b^3)^{9}\)
    3. the last \(3\) terms in the binomial expansion of \((3y^2-x^2)^{7}\)
    4. the first \(3\) terms in the binomial expansion of \(\left(\dfrac{x}{y}-\dfrac{y}{x}\right)^{10}\)
    5. the last \(4\) terms in the binomial expansion of \(\left(m^3n+\dfrac{1}{2}n^2\right)^{6}\)
    Answer
    1. \(x^{5} y^{5}-20 x^{5} y^{4}+160 x^{5} y^{3}\)
    2. \(512 a^{18}+2304 a^{16} b^{3}\)
    3. \(-189 x^{10} y^{4}+21 x^{12} y^{2}-x^{14}\)
    4. \(\dfrac{x^{10}}{y^{10}}-10 \dfrac{x^{8}}{y^{8}}+45 \dfrac{x^{6}}{y^{6}}\)
    5. \(\dfrac{5}{2} m^{9} n^{9}+\dfrac{15}{16} m^{6} n^{10}+\dfrac{3}{16} m^{3} n^{11}+\dfrac{1}{64} n^{12}\)

    Exercise \(\PageIndex{5}\)

    Determine:

    1. the \(5\)th term in the binomial expansion of \((x+y)^{7}\)
    2. the \(3\)rd term in the binomial expansion of \((x^2-y)^{9}\)
    3. the \(10\)th term in the binomial expansion of \((2-w)^{11}\)
    4. the \(5\)th term in the binomial expansion of \((2x+xy)^{7}\)
    5. the \(7\)th term in the binomial expansion of \((2a+5b)^{6}\)
    6. the \(6\)th term in the binomial expansion of \((3p^2-q^3p)^{7}\)
    7. the \(10\)th term in the binomial expansion of \(\left(4+\dfrac{b}{2}\right)^{13}\)
    Answer
    1. \(35 x^{3} y^{4}\)
    2. \(36 x^{14} y^{2}\)
    3. \(-220 w^{9}\)
    4. \(280 x^{7} y^{4}\)
    5. \(15625 b^{6}\)
    6. \(-189 p^{9} q^{15}\)
    7. \(\dfrac{715}{2} b^{9}\)

    Exercise \(\PageIndex{6}\)

    Determine:

    1. the \(x^3y^{6}\)-term in the binomial expansion of \((x+y)^{9}\)
    2. the \(r^4s^4\)-term in the binomial expansion of \((r^2-s)^{6}\)
    3. the \(x^{4}\)-term in the binomial expansion of \((x-1)^{11}\)
    4. the \(x^3y^{6}\)-term in the binomial expansion of \((x^3+5y^2)^{4}\)
    5. the \(x^{7}\)-term in the binomial expansion of \((2x-x^2)^{5}\)
    6. the imaginary part of the number \((1+i)^3\)
    Answer
    1. \(84 x^{3} y^{6}\)
    2. \(15 r^{4} s^{4}\)
    3. \(-330 x^{4}\)
    4. \(500 x^{3} y^{6}\)
    5. \(80 x^{7}\)
    6. \(2 i\)

    This page titled 25.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.