Soal kombinatorik (1)


Berikut ini soal-soal dan pembahasan kombinatorik yang dapat digunakan untuk menambah wawasan tentang konsep -konsep kombinatorik.
1. Buktikan bahwa _{n+1}C_{r}= _{n}C_{r-1}+_{n}C_{r}

Bukti :

kita gunakan rumus kombinasi _{n}C_{r}= \frac{n!}{(n-r)!.r!}
Sehingga :

_{n}C_{r-1}= \frac{n!}{(n-(r-1))!.(r-1)!}

_{n}C_{r-1}= \frac{n!}{(n+1-r)!.(r-1)!}

_{n}C_{r-1}= \frac{n!.r}{(n+1-r)(n-r)!.r.(r-1)!}

_{n}C_{r-1}= \frac{r}{(n+1-r)}.\frac{n!}{(n-r)!.r!}

sehingga :
_{n}C_{r-1}+_{n}C_{r}= \frac{r}{(n+1-r)}.\frac{n!}{(n-r)!.r!}+\frac{n!}{(n-r)!.r!}

_{n}C_{r-1}+_{n}C_{r}= \left(\frac{r}{(n+1-r)}+1\right).\frac{n!}{(n-r)!.r!}

_{n}C_{r-1}+_{n}C_{r}= \left(\frac{r+n+1-r}{(n+1-r)}\right).\frac{n!}{(n-r)!.r!}

_{n}C_{r-1}+_{n}C_{r}= \left(\frac{n+1}{(n+1-r)}\right).\frac{n!}{(n-r)!.r!}

_{n}C_{r-1}+_{n}C_{r}= \frac{(n+1).n!}{(n+1-r).(n-r)!.r!}

_{n}C_{r-1}+_{n}C_{r}= \frac{(n+1)!}{(n+1-r)!.r!}

_{n}C_{r-1}+_{n}C_{r}=_{n+1}C_{r}

2. Buktikan bahwa _{n}C_{r+1}= \frac{n-r}{r+1}._{n}C_{r}

Bukti :
_{n}C_{r+1}=\frac{n!}{(n-(r+1))!.(r+1)!}

=\frac{n!}{((n-1)-r)!.(r+1).r!}

=\frac{n!}{((n-1)-r)!.(r+1).r!}.\frac{n-r}{n-r}

= \frac{(n-r).n!}{(r+1).(n-r).((n-1)-r)!.r!}

=\frac{n-r}{r+1}.\frac{n!}{(n-r).((n-1)-r)!.r!}

=\frac{n-r}{r+1}.\frac{n!}{(n-r)!.r!}

=\frac{n-r}{r+1}._{n}C_{r}

Jadi terbukti bahwa _{n}C_{r+1}= \frac{n-r}{r+1}._{n}C_{r}

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