# 25.3: Exercises

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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. $$\dbinom{16}{5}$$
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. $$(-p-q)^5$$
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. $$(1-i)^3$$
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 $$\left(m^3n+\dfrac{1}{2}n^2\right)^{6}$$
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 $$\left(4+\dfrac{b}{2}\right)^{13}$$
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 $$(1+i)^3$$
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. $$2 i$$