Contoh Soal 3 Kaidah Pencacahan

 $\begin{array}{ll}11.& \text{Berikut ini nilainya tidak sama dengan}\\ & C(7,5)\text{adalah}....\\ \\ & (i){\displaystyle \frac{7!}{5!(7-5)!}}\\ \\ & (ii)C(6,1)\\ \\ & (iii){\displaystyle \frac{P(7,5)}{5!}}\\ \\ & (iv)\left(\begin{array}{c}6\\ 1\end{array}\right)\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle (i),(ii),\mathrm{\&}(iii)}& & & \text{d}.& {\displaystyle \text{hanya}(i)}\\ \\ \text{b}.& {\displaystyle (i)\mathrm{\&}(iii)}& \text{c}.& (ii)\mathrm{\&}(iv)& \text{e}.& {\displaystyle \text{hanya}(iv)}\end{array}\\ \\ & \text{Jawab}:\\ & C(7,5)={\displaystyle \frac{P(7,5)}{5!}=\frac{7!}{5!(7-5)!}}\end{array}$

$\begin{array}{ll}12.& \text{Nilai}n\text{yang memenuhi persamaan}\\ & \left(\begin{array}{c}100\\ 45\end{array}\right)=\left(\begin{array}{c}100\\ 5n\end{array}\right)\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 15}& & & \text{d}.& {\displaystyle 12}\\ \\ \text{b}.& {\displaystyle 14}& \text{c}.& {\displaystyle 13}& \text{e}.& {\displaystyle 11}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Diketahui bahwa}\\ & \left(\begin{array}{c}100\\ 45\end{array}\right)=\left(\begin{array}{c}100\\ 5n\end{array}\right),\text{maka}\\ & 45+5n=100\\ & 5n=100-45=55\\ & n={\displaystyle \frac{55}{5}=11}\end{array}\end{array}$

$\begin{array}{ll}13.& \text{Koefisien suku ke-4 dari}(2x-3{)}^4\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle -216}& & & \text{d}.& {\displaystyle 81}\\ \\ \text{b}.& {\displaystyle -96}& \text{c}.& {\displaystyle 16}& \text{e}.& {\displaystyle 216}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Diketahui bahwa}\\ & (2x-3{)}^4={\displaystyle \sum _{i=0}^4\left(\begin{array}{c}4\\ i\end{array}\right)(2x{)}^{4-i}(-3{)}^i}\\ & \text{Suku ke-4-nya adalah}:r=4.\\ & \text{Suku ke-r}=\left(\begin{array}{c}n\\ r-1\end{array}\right)a^{n-r+1}{b}^{r-1}\\ & \text{Sehingga suku ke-4 adalah}:\\ & =\left(\begin{array}{c}4\\ 4-1\end{array}\right)(2x{)}^{4-4+1}(-3{)}^{4-1}\\ & =\left(\begin{array}{c}4\\ 3\end{array}\right)(2x{)}^1(-3{)}^3\\ & ={\displaystyle \frac{4!}{3!\times 1!}2x(-27)}\\ & =-4.2.27x\\ & =-216\end{array}\end{array}$

$\begin{array}{ll}14.& \text{Bentuk sederhana dari}{\displaystyle \sum _{r=1}^{n}r{\displaystyle \left(\begin{array}{c}n\\ r\end{array}\right)}}\\ & \text{dengan}\left(\begin{array}{c}n\\ r\end{array}\right)={\displaystyle \frac{n!}{r!(n-r)!}\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 2^{n+1}}& & & \text{d}.& {\displaystyle 3^n}\\ \\ \text{b}.& {\displaystyle n{2}^{n-1}}& \text{c}.& {\displaystyle n{2}^n}& \text{e}.& {\displaystyle 3^{n+1}}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}{\displaystyle \sum _{r=1}^{n}r{\displaystyle \left(\begin{array}{c}n\\ r\end{array}\right)}}& ={\displaystyle \sum _{r=1}^{n}r{\displaystyle \frac{n!}{r!(n-r)!}}}\\ & ={\displaystyle \sum _{r=1}^{n}r{\displaystyle \frac{n(n-1)!}{r(r-1)!(n-r)!}}}\\ & =n{\displaystyle \sum _{r=1}^n{\displaystyle \frac{(n-1)!}{(r-1)!(n-r)!}}}\\ & =n{\displaystyle \sum _{r=1}^n{\displaystyle \frac{(n-1)!}{(r-1)!((n-1)-(r-1))!}}}\\ & ={\displaystyle \sum _{r=1}^n\left(\begin{array}{c}n-1\\ r-1\end{array}\right)}\\ & =n{.2}^{r-1}\end{array}\end{array}$

$\begin{array}{ll}15.& \text{Banyaknya diagonal segi 6 adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 15}& & & \text{d}.& {\displaystyle 9}\\ \\ \text{b}.& {\displaystyle 14}& \text{c}.& {\displaystyle 10}& \text{e}.& {\displaystyle 6}\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}& \text{Banyak diagonal segi}-n\text{adalah}:\\ & C(n,2)-n.\text{Jika seperti soal dengan}\\ & n=6,\text{maka}\\ & C(6,2)={\displaystyle \frac{6!}{2!\times 4!}=\frac{6\times 5\times \text{⧸}4!}{2\times 1\times \text{⧸}4!}=15}\\ & \text{Sehingga}\\ & C(6,2)-6=15-6=9\end{array}\end{array}$