Factorial Expression : Exercise

Factorials

We define $\boldsymbol{n} !=\boldsymbol{n}(\boldsymbol{n}-\mathbf{1})(\boldsymbol{n}-\mathbf{2}) \cdots \mathbf{3} \cdot \mathbf{2} \cdot \mathbf{1}$ if $n$ is a nonnegative integer.

An empty product is normally defined to be 1 .

With this convention, $0 !=1$

An alternative is to define $\boldsymbol{n} !$ recursively on the nonnegative integers.

$\boldsymbol{n} != \begin{cases}1 & \text { if } \boldsymbol{n}=\mathbf{0} \\ \boldsymbol{n}(\boldsymbol{n}-\mathbf{1}) ! & \text { if } \boldsymbol{n} \geq \mathbf{1}\end{cases}$

Exercise

  1. Evaluate
    $\begin{array}{lll} \text{(a)}\ 2 !& \text{(b)}\ 3 !& \text{(c)}\ 4 !\\\\ \text{(e)}\ 5 !& \text{(f)}\ 6 !& \text{(g)}\ 10 ! \end{array}$


  2. $\begin{aligned} \text{(a)}\ &2 !=2 \times 1=2 \\\\ \text{(b)}\ &3 !=3 \times 2 \times 1=6 \\\\ \text{(c)}\ &4 !=4 \times 3 \times 2 \times 1=24 \\\\ \text{(d)}\ &5 !=5 \times 4 \times 3 \times 2 \times 1=120 \\\\ \text{(e)}\ &6 !=6 \times 5 \times 4 \times 3 \times 2 \times 1=720 \\\\ \text{(f)}\ &10 !=10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=362880 \end{aligned}$

  3. Express in factorial form:
    $\begin{array}{ll} \text{(a)}\ 4 \times 3 \times 2 \times 1 & \text{(b)}\ 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \quad \\\\ \text{(c)}\ 6 \times 5& \text{(d)}\ 8 \times 7 \times 6\\\\ \text{(e)}\ 10 \times 9 \times 8 \times 7& \text{(f)}\ 15 \times 14 \times 13 \times 12\\\\ \text{(g)}\ \dfrac{9 \times 8 \times 7}{3 \times 2 \times 1}& \text{(h)}\ \dfrac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1}\\\\ \text{(i)}\ \dfrac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} \end{array}$


  4. $\begin{aligned} \text{(a)}\ &\quad 4 \times 3 \times 2 \times 1\\\\ &=4 !\\\\ \text{(b)}\ &\quad 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\\\\ &=7 !\\\\ \text{(c)}\ &\quad 6 \times 5\\\\ &=\dfrac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{6 !}{4 !}\\\\ \text{(d)}\ &\quad 8 \times 7 \times 6\\\\ &=\dfrac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{8 !}{5 !}\\\\ \text{(e)}\ &\quad 10 \times 9 \times 8 \times 7\\\\ &=\dfrac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{10 !}{6 !}\\\\ \text{(f)}\ &\quad 15 \times 14 \times 13 \times 12\\\\ &=\dfrac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{15 !}{11 !}\\\\ \text{(g)}\ &\quad \dfrac{9 \times 8 \times 7}{3 \times 2 \times 1}\\\\ &=\dfrac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(6 \times 5 \times 4 \times 3 \times 2 \times 1)}\\\\ &=\dfrac{9 !}{3 ! 6 !}\\\\ \text{(h)}\ &\quad \dfrac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1}\\\\\ &=\dfrac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1)(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}\\\\ &=\dfrac{13 !}{4 ! 9 !}\\\\ \text{(i)}\ &\quad \dfrac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(5 \times 4 \times 3 \times 2 \times 1)( 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}\\\\ &=\dfrac{15 !}{5 ! 10 !} \end{aligned}$

    $\begin{aligned} &\textbf{Alternative Method }\\\\ \text{(a)}\ &\quad 4 \times 3 \times 2 \times 1\\\\ &=4 !\\\\ \text{(b)}\ &\quad 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\\\\ &=7 !\\\\ \text{(c)}\ &\quad 6 \times 5=\dfrac{6 \times 5 \times 4 !}{4 !}\\\\ &=\dfrac{6 !}{4 !}\\\\ \text{(d)}\ &\quad 8 \times 7 \times 6\\\\ &=\dfrac{8 \times 7 \times 6 \times 5 !}{5 !}\\\\ &=\dfrac{8 !}{5 !}\\\\ \text{(e)}\ &\quad 10 \times 9 \times 8 \times 7\\\\ &=\dfrac{10 \times 9 \times 8 \times 7 \times 6 !}{6 !}\\\\ &=\dfrac{10 !}{6 !}\\\\ \text{(f)}\ &\quad 15 \times 14 \times 13 \times 12=\dfrac{15 \times 14 \times 13 \times 12 \times 11 !}{11 !}\\\\ &=\dfrac{15 !}{11 !}\\\\ \text{(g)}\ &\quad \dfrac{9 \times 8 \times 7}{3 \times 2 \times 1}\\\\ &=\dfrac{9 \times 8 \times 7 \times 6 !}{3 \times 2 \times 1 \times 6 !}\\\\ &=\dfrac{9 !}{3 ! 6 !}\\\\ \text{(h)}\ &\quad \dfrac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{13 \times 12 \times 11 \times 10 \times 9 !}{4 \times 3 \times 2 \times 1 \times 9 !}\\\\ &=\dfrac{13 !}{4 ! 9 !} \\\\ \text{(i)}\ &\quad \dfrac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1}\\\\ &=\dfrac{15 \times 14 \times 13 \times 12 \times 11 \times 10 !}{5 \times 4 \times 3 \times 2 \times 1 \times 10 !}\\\\ &=\dfrac{15 !}{5 ! 10 !} \end{aligned}$

  5. Simplify without using a calculator:
    $\begin{array}{ll} \text{(a)}\ \dfrac{7 !}{6 !}& \text{(b)}\ \dfrac{8 !}{6 !}\\\\ \text{(c)}\ \dfrac{12 !}{10 !}& \text{(d)}\ \dfrac{120 !}{119 !}\\\\ \text{(e)}\ \dfrac{10 !}{8 ! \times 2 !}& \text{(f)}\ \dfrac{100 !}{98 ! \times 2 !}\\\\ \text{(g)}\ \dfrac{7 !}{3 !}& \text{(h)}\ \dfrac{8 !}{5 !}\\\\ \text{(i)}\ \dfrac{4 !}{2 ! 2 !}& \text{(j)}\ \dfrac{6 !}{3 ! 2 !}\\\\ \text{(k)}\ \dfrac{6 !}{(3 !)^{2}}& \text{(l)}\ \dfrac{5 !}{3 !} \times \dfrac{7 !}{4 !} \end{array}$


  6. $\begin{array}{l} \text{(a)}\ \dfrac{7 !}{6 !}=\dfrac{7 \times 6 !}{6 !}=7\\\\ \text{(b)}\ \dfrac{8 !}{6 !}=\dfrac{8 \times 7 \times 6 !}{6 !}=56\\\\ \text{(c)}\ \dfrac{12 !}{10 !}=\dfrac{12 \times 11 \times 10 !}{10 !}=132\\\\ \text{(d)}\ \dfrac{120 !}{119 !}=\dfrac{120 \times 119 !}{119 !}=120\\\\ \text{(e)}\ \dfrac{10 !}{8 ! \times 2 !}=\dfrac{10 \times 9 \times 8 !}{8 ! \times(2 \times 1)}=45\\\\ \text{(f)}\ \dfrac{100 !}{98 ! \times 2 !}=\dfrac{100 \times 99 \times 98 !}{98 ! \times 2 \times 1}=4950\\\\ \text{(g)}\ \dfrac{7 !}{3 !}=\dfrac{7 \times 6 \times 5 \times 4 \times 3 !}{3 !}=840\\\\ \text{(h)}\ \dfrac{8 !}{5 !}=\dfrac{8 \times 7 \times 6 \times 5 !}{5 !}=336\\\\ \text{(i)}\ \dfrac{4 !}{2 ! 2 !}=\dfrac{4 \times 3 \times 2 !}{2 !(2 \times 1)}=6\\\\ \text{(j)}\ \dfrac{6 !}{3 ! 2 !}=\dfrac{6 \times 5 \times 4 \times 3 !}{3 ! \times(2 \times 1)}=60\\\\ \text{(k)}\ \dfrac{6 !}{(3 !)^{2}}=\dfrac{6 \times 5 \times 4 \times 3 !}{3 !(3 \times 2 \times 1)}=20\\\\ \text{(l)}\ \dfrac{5 !}{3 !} \times \dfrac{7 !}{4 !}=(5 \times 4) \times(7 \times 6 \times 5)=4200 \end{array}$

  7. Simplify:
    $\begin{array}{l} \text{(a)}\ \dfrac{n !}{(n-1) !}\\\\ \text{(b)}\ \dfrac{(n+2) !}{n !}\\\\ \text{(c)}\ \dfrac{(n+1) !}{(n-1) !} \end{array}$


  8. $\begin{aligned} \text{(a)}\ &\dfrac{n !}{(n-1) !}\\\\\ &=\dfrac{n(n-1) !}{(n-1) !}\\\\ &=n \\\\ \text{(b)}\ &\dfrac{(n+2) !}{n !}\\\\ &=\dfrac{(n+2)(n+1) n !}{n !}\\\\ &=n^{2}+3 n+2 \\\\ \text{(c)}\ &\dfrac{(n+1) !}{(n-1) !}\\\\ &=\dfrac{(n+1) n(n-1) !}{(n-1) !}\\\\ &=n^{2}+n \end{aligned}$

  9. Rewrite each of the following using factorial notation.
    $\begin{array}{l} \text{(a)}\ n(n-1)(n-2)(n-3)\\\\ \text{(b)}\ n(n-1)(n-2)(n-3)(n-4)(n-5)\\\\ \text{(c)}\ \dfrac{n(n-1)(n-2)}{5 \times 4 \times 3 \times 2 \times 1}\\\\ \text{(d)}\ \dfrac{n(n-1)(n-2)(n-3)(n-4)}{3 \times 2 \times 1} \end{array}$


  10. $\begin{aligned} \text{(a)}\ &n(n-1)(n-2)(n-3)\\\\ &=\frac{n(n-1)(n-2)(n-3)(n-4) !}{(n-4) !}\\\\ &=\frac{n !}{(n-4) !} \\\\ \text{(b)}\ &n(n-1)(n-2)(n-3)(n-4)(n-5)\\\\ &=\frac{n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6) !}{(n-6) !}\\\\ &=\frac{n !}{(n-6) !} \\\\ \text{(c)}\ &\frac{n(n-1)(n-2)}{5 \times 4 \times 3 \times 2 \times 1}\\\\ &=\frac{n(n-1)(n-2)(n-3) !}{5 !(n-3) !}=\frac{n !}{5 !(n-3) !} \\\\ \text{(d)}\ &\frac{n(n-1)(n-2)(n-3)(n-4)}{3 \times 2 \times 1}\\\\ &=\frac{n(n-1)(n-2)(n-3)(n-4)(n-5) !}{3 !(n-5) !}\\\\ &=\frac{n !}{3 !(n-5) !} \end{aligned}$

  11. Express the following as a single factorial notation.
    (a) $n !(n+1)$
    (b) $(n-1) !\left(n^{2}+n\right)$
    (c) $(n+4)(n+5)(n+3) !$
    (d) $n !\left(n^{2}+3 n+2\right)$
    (e) $(n+1)(n+2)(n+3)$
    (f) $(n-3)(n-4)(n-5)$


  12. $\begin{aligned} \text { (a) } & n !(n+1) \\\\ &=(n+1) n ! \\\\ &=(n+1) ! \\\\ \text { (b) } &(n-1) !\left(n^{2}+n\right) \\\\ &=\left(n^{2}+n\right)(n-1) ! \\\\ &=(n+1) n(n-1) ! \\\\ &=(n+1) ! \\\\ &=(n+5)(n+4)(n+3) ! \\\\ &=(n+5) !\\\\ \text { (c) } &(n+4)(n+5)(n+3) ! \\\\ &=(n+5)(n+4)(n+3) ! \\\\ &=(n+5) !\\\\ \text { (d) } & n !\left(n^{2}+3 n+2\right) \\\\ =&(n+2)(n+1) n ! \\\\ =&(n+2) !\\\\ \text { (e) } &(n+1)(n+2)(n+3) \\\\ =& \frac{(n+3)(n+2)(n+1) n !}{n !} \\\\ =& \frac{(n+3) !}{n !}\\\\ \text { (f) } & (n-3)(n-4)(n-5) \\\\ &=\frac{(n-3)(n-4)(n-5)(n-6) !}{(n-6) !} \\\\ &=\frac{(n-3) !}{(n-6) !} \end{aligned}$

  13. Write as a product by factorizing:
    (a) $5 !+4 !$
    (b) $11 !-10 !$
    (c) $5 !+7 !$
    (d) $12 !-10 !$
    (e) $9 !+8 !+7 !$
    (f) $7 !-6 !+8 !$
    (g) $12 !-2 \times 11 !$
    (h) $3 \times 9 !+5 \times 8 !$


  14. $\begin{aligned} \text { (a) } & \quad 5 !+4 ! \\\\ &= 5 \times 4 !+4 ! \\\\ &=(5+1) 4 ! \\\\ &= 6 \times 4 ! \\\\ \text { (b) } & \quad 11 !-10 ! \\\\ &=(11-1) 10 ! \\\\ &= 10 \times 10 !\\\\ \text { (c) } & \quad 5 !+7 ! \\\\ &= 5 !+7 \times 6 \times 5 ! \\\\ &=(1+42) 5 ! \\\\ &= 43 \times 5 ! \\\\ \text { (d) } & \quad 12 !-10 ! \\\\ &= 12 \times 11 \times 10 !-10 ! \\\\ &=(132-1) 10 ! \\\\ &= 131 \times 10 !\\\\ \text { (e) } & \quad 9 !+8 !+7 ! \\\\ &= 9 \times 8 \times 7 !+8 \times 7 !+7 ! \\\\ &=(72+8+1) 7 ! \\\\ &= 81 \times 7 ! \\\\ \text { (f) } & \quad 7 !-6 !+8 ! \\\\ &= 7 \times 6 !-6 !+8 \times 7 \times 6 ! \\\\ &=(7-1+56) 6 ! \\\\ &= 62 \times 6 !\\\\ \text { (g) } & \quad 12 !-2 \times 11 ! \\\\ &= 12 \times 11 !-2 \times 11 ! \\\\ &=(12-2) 11 ! \\\\ &= 10 \times 11 ! \\\\ \text { (h) } & \quad 3 \times 9 !+5 \times 8 ! \\\\ &= 3 \times 9 \times 8 !+5 \times 8 ! \\\\ &=(27+5) 8 ! \\\\ &= 32 \times 8 ! \end{aligned}$

  15. Simplify by factorizing:
    $\begin{array}{l} \text{(a)}\ \dfrac{12 !-11 !}{11}\\\\ \text{(b)}\ \dfrac{10 !+9 !}{11}\\\\ \text{(c)}\ \dfrac{10 !-8 !}{89}\\\\ \text{(d)}\ \dfrac{10 !-9 !}{9 !}\\\\ \text{(e)}\ \dfrac{6 !+5 !-4 !}{4 !}\\\\ \text{(f)}\ \dfrac{n !+(n-1) !}{(n-1) !}\\\\ \text{(g)}\ \dfrac{n !-(n-1) !}{n-1}\\\\ \text{(h)}\ \dfrac{(n+2) !+(n+1) !}{n+3} \end{array}$


  16. $\begin{aligned} \text { (a) } & \quad \dfrac{12 !-11 !}{11} \\\\ &= \dfrac{12 \times 11 !-11 !}{11} \\\\ &= \dfrac{(12-1) 11 !}{11} \\\\ &= \dfrac{11 \times 11 !}{11} \\\\ &= 11 !\\\\ \text { (b) } & \quad \dfrac{10 !+9 !}{11} \\\\ &= \dfrac{10 \times 9 !+9 !}{11} \\\\ &= \dfrac{(10+1) 9 !}{11} \\\\ &= \dfrac{11 \times 9 !}{11} \\\\ &= 9 !\\\\ \text { (c) } & \quad \dfrac{10 !-8 !}{89} \\\\ &= \dfrac{10 \times 9 \times 8 !-8 !}{89} \\\\ &= \dfrac{(90-1) \times 8 !}{89} \\\\ &= \dfrac{89 \times 8 !}{89} \\\\ &= 8 !\\\\ \text { (d) } & \quad \dfrac{10 !-9 !}{9} \\\\ &= \dfrac{10 \times 9 !-9 !}{9} \\\\ &= \dfrac{(10-1) 9 !}{9} \\\\ &= \dfrac{9 \times 9 !}{9} \\\\ &= 9 !\\\\ \text { (e) } & \quad \dfrac{6 !+5 !-4 !}{4 !} \\\\ &= \dfrac{6 \times 5 \times 4 !+5 \times 4 !-4 !}{4 !} \\\\ &= \dfrac{(30+5-1) 4 !}{4 !} \\\\ &= 34\\\\ \text { (f) } & \quad \dfrac{n !+(n-1) !}{(n-1) !} \\\\ &=\dfrac{n(n-1) !+(n-1) !}{(n-1) !} \\\\ &=\dfrac{(n+1)(n-1) !}{(n-1) !} \\\\ &=n+1\\\\ \text { (g) } & \quad \dfrac{n !-(n-1) !}{n-1} \\\\ &=\dfrac{n(n-1) !-(n-1) !}{n-1} \\\\ &=\dfrac{(n-1)(n-1) !}{n-1} \\\\ &=(n-1) !\\\\ \text { (h) } & \quad \dfrac{(n+2) !+(n+1) !}{n+3} \\\\ &= \dfrac{(n+2)(n+1) !+(n+1) !}{n+3} \\\\ &= \dfrac{(n+2+1)(n+1) !}{n+3} \\\\ &= \dfrac{(n+3)(n+1) !}{n+3} \\\\ &=(n+1) ! \end{aligned}$