Question

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

Answer 1

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

$\displaystyle\sum_{r=4}^8{}_{13}C_r={}_{13}C_4+\cdots+{}_{13}C_8=6721$

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

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

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