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=(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=(13!)/(4!(13-4)!)=(13!)/(4!9!)`

`=(13*12*\cdots*6*5)/(9*8*\cdots*2*1)`

`=(13*12*11*10)/(4*3*2*1)`

`=13*11*5`

`=715`

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