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Buktikan kebenaran sifat berikut! (n+1)​Cr​=n​C(r−1)​+n​Cr​

Pertanyaan

Buktikan kebenaran sifat berikut!

straight C presubscript open parentheses straight n plus 1 close parentheses end presubscript subscript straight r equals straight C presubscript straight n subscript open parentheses straight r minus 1 close parentheses end subscript plus straight C presubscript straight n subscript straight r

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

terbukti bahwa C presubscript open parentheses n plus 1 close parentheses end presubscript subscript r equals C presubscript n subscript open parentheses r minus 1 close parentheses end subscript plus C presubscript n subscript r.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah terbukti bahwa C presubscript open parentheses n plus 1 close parentheses end presubscript subscript r equals C presubscript n subscript open parentheses r minus 1 close parentheses end subscript plus C presubscript n subscript r.

Kombinasi r unsur dari n unsur berbeda adalah banyak susunan berbeda dari r unsur berbeda yang diambil dari n unsur berbeda tanpa memperhatikan urutan. Kombinasi dirumuskan sebagai berikut.

C presubscript n subscript r equals fraction numerator n factorial over denominator r factorial times open parentheses n minus r close parentheses factorial end fraction

dengan n dan r bilangan asli.

Akan dibuktikan kebenaran dari C presubscript open parentheses n plus 1 close parentheses end presubscript subscript r equals C presubscript n subscript open parentheses r minus 1 close parentheses end subscript plus C presubscript n subscript r

Bukti:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell C presubscript open parentheses n plus 1 close parentheses end presubscript subscript r end cell equals cell C presubscript n subscript open parentheses r minus 1 close parentheses end subscript plus C presubscript n subscript r end cell row cell fraction numerator open parentheses n plus 1 close parentheses factorial over denominator r factorial times open parentheses n plus 1 minus r close parentheses factorial end fraction end cell equals cell fraction numerator n factorial over denominator open parentheses r minus 1 close parentheses factorial times open parentheses n minus open parentheses r minus 1 close parentheses close parentheses factorial end fraction plus fraction numerator n factorial over denominator r factorial times open parentheses n minus r close parentheses factorial end fraction end cell row cell fraction numerator open parentheses n plus 1 close parentheses times n factorial over denominator r times open parentheses r minus 1 close parentheses factorial times open parentheses n minus r plus 1 close parentheses factorial end fraction end cell equals cell fraction numerator n factorial over denominator open parentheses r minus 1 close parentheses factorial times open parentheses n minus r plus 1 close parentheses factorial end fraction plus fraction numerator n factorial over denominator r times open parentheses r minus 1 close parentheses factorial times open parentheses n minus r close parentheses factorial end fraction end cell row cell fraction numerator open parentheses n plus 1 close parentheses over denominator r times open parentheses r minus 1 close parentheses factorial times open parentheses n minus r plus 1 close parentheses factorial end fraction end cell equals cell fraction numerator 1 over denominator open parentheses r minus 1 close parentheses factorial times open parentheses n minus r plus 1 close parentheses factorial end fraction plus fraction numerator 1 over denominator r times open parentheses r minus 1 close parentheses factorial times open parentheses n minus r close parentheses factorial end fraction end cell row cell fraction numerator open parentheses n plus 1 close parentheses over denominator r times open parentheses r minus 1 close parentheses factorial times open parentheses n minus r plus 1 close parentheses factorial end fraction end cell equals cell fraction numerator r times open parentheses n minus r close parentheses factorial plus open parentheses n minus r plus 1 close parentheses factorial over denominator r times open parentheses r minus 1 close parentheses factorial times open parentheses n minus r plus 1 close parentheses factorial times open parentheses n minus r close parentheses factorial end fraction end cell row cell n plus 1 end cell equals cell fraction numerator r times open parentheses n minus r close parentheses factorial plus open parentheses n minus r plus 1 close parentheses open parentheses n minus r close parentheses factorial over denominator open parentheses n minus r close parentheses factorial end fraction end cell row cell n plus 1 end cell equals cell fraction numerator open parentheses r plus open parentheses n minus r plus 1 close parentheses close parentheses open parentheses n minus r close parentheses factorial over denominator open parentheses n minus r close parentheses factorial end fraction end cell row cell n plus 1 end cell equals cell n plus 1 end cell end table end style

Dengan demikian, terbukti bahwa C presubscript open parentheses n plus 1 close parentheses end presubscript subscript r equals C presubscript n subscript open parentheses r minus 1 close parentheses end subscript plus C presubscript n subscript r.

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