Tunjukkan bahwa C(11, 3)=C(11, 8)=3!P(11, 3)​  !

Pertanyaan

Tunjukkan bahwa C left parenthesis 11 comma space 3 right parenthesis equals C left parenthesis 11 comma space 8 right parenthesis equals fraction numerator P left parenthesis 11 comma space 3 right parenthesis over denominator 3 factorial end fraction space space

S. Nur

Master Teacher

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Jawaban

telah ditunjukkan bahwa C left parenthesis 11 comma space 3 right parenthesis equals C left parenthesis 11 comma space 8 right parenthesis equals fraction numerator P left parenthesis 11 comma space 3 right parenthesis over denominator 3 factorial end fraction space space.

Pembahasan

Ingat rumus kombinasi dan permutasi berikut.

C open parentheses a comma b close parentheses equals fraction numerator a factorial over denominator b factorial left parenthesis a minus b right parenthesis factorial end fraction

P left parenthesis a comma b right parenthesis equals fraction numerator a factorial over denominator left parenthesis a minus b factorial right parenthesis end fraction

Berdasarkan rumus di atas, maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell C left parenthesis 11 comma 3 right parenthesis end cell equals cell fraction numerator 11 factorial over denominator 3 factorial left parenthesis 11 minus 3 right parenthesis factorial end fraction end cell row blank equals cell fraction numerator 11 factorial over denominator 3 factorial 8 factorial end fraction end cell row blank equals cell fraction numerator 11 factorial over denominator left parenthesis 11 minus 8 right parenthesis factorial 8 factorial end fraction end cell row blank equals cell C left parenthesis 11 comma 8 right parenthesis end cell end table
 

table attributes columnalign right center left columnspacing 0px end attributes row cell C left parenthesis 11 comma 8 right parenthesis end cell equals cell fraction numerator 11 factorial over denominator 8 factorial left parenthesis 11 minus 8 right parenthesis factorial end fraction end cell row blank equals cell fraction numerator 11 factorial over denominator 8 factorial 3 factorial end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 11 factorial over denominator 8 factorial end fraction end style close parentheses over denominator 3 factorial end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 11 factorial over denominator left parenthesis 11 minus 3 right parenthesis factorial end fraction end style close parentheses over denominator 3 factorial end fraction end cell row blank equals cell fraction numerator P left parenthesis 11 comma 3 right parenthesis over denominator 3 factorial end fraction end cell end table

Dengan demikian, telah ditunjukkan bahwa C left parenthesis 11 comma space 3 right parenthesis equals C left parenthesis 11 comma space 8 right parenthesis equals fraction numerator P left parenthesis 11 comma space 3 right parenthesis over denominator 3 factorial end fraction space space.

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