Mathematics

Question

if a given set has thirteen elements how many subsets have somewhere from four through eight elements

1 Answer

  • Of [tex]n[/tex] elements, there are [tex]{}_nC_r=\dfrac{n!}{r!(n-r)!}[/tex] ways of choosing any [tex]r[/tex] elements. So the number of subsets that can be chosen from the set of 13 elements, each consisting of 4 to 8 elements, is

    [tex]\displaystyle\sum_{r=4}^8{}_{13}C_r={}_{13}C_4+\cdots+{}_{13}C_8=6721[/tex]

    To compute the actual numbers, you have, for example,

    [tex]{}_{13}C_4=\dfrac{13!}{4!(13-4)!}=\dfrac{13!}{4!9!}[/tex]
    [tex]=\dfrac{13\times12\times\cdots\times6\times5}{9\times8\times\cdots\times2\times1}[/tex]
    [tex]=\dfrac{13\times12\times11\times10}{4\times3\times2\times1}[/tex]
    [tex]=13\times11\times5[/tex]
    [tex]=715[/tex]

    so there are 715 ways of picking subsets of size 4. Compute the others similarly, then add them up.
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