Exercise 24 Page 704

Now, because the order is not important in a combination, we will cross out any duplicate quadruplets. Keep in mind that any outcome with the same letters, regardless of the order, is the same quadruplet. For example, DEFG and DEGF are the same quadruplet. \begin{gathered} {\color{#A800DD}{\text{DEFG}}} \quad {\color{#FD9000}{\text{DEFH}}} \quad {\color{#A800DD}{\cancel\text{DEGF}}} \quad {\color{#FF0000}{\text{DEGH}}} \quad {\color{#FD9000}{\cancel\text{DEHF}}} \quad {\color{#FF0000}{\cancel\text{DEHG}}}\\ {\color{#A800DD}{\cancel\text{DFEG}}} \quad {\color{#FD9000}{\cancel\text{DFEH}}} \quad {\color{#A800DD}{\cancel\text{DFGE}}} \quad {\color{#009600}{\text{DFGH}}} \quad {\color{#FD9000}{\cancel\text{DFHE}}} \quad {\color{#009600}{\cancel\text{DFHG}}}\\ {\color{#A800DD}{\cancel\text{DGEF}}} \quad {\color{#FF0000}{\cancel\text{DGEH}}} \quad {\color{#A800DD}{\cancel\text{DGFE}}} \quad {\color{#009600}{\cancel\text{DGFH}}} \quad {\color{#FF0000}{\cancel\text{DGHE}}} \quad {\color{#009600}{\cancel\text{DGHF}}}\\ {\color{#FD9000}{\cancel\text{DHEF}}} \quad {\color{#FF0000}{\cancel\text{DHEG}}} \quad {\color{#FD9000}{\cancel\text{DHFE}}} \quad {\color{#009600}{\cancel\text{DHFG}}} \quad {\color{#FF0000}{\cancel\text{DHGE}}} \quad {\color{#009600}{\cancel\text{DHGF}}}\\[0.8em] {\color{#A800DD}{\cancel\text{EDFG}}} \quad {\color{#FD9000}{\cancel\text{EDFH}}} \quad {\color{#A800DD}{\cancel\text{EDGF}}} \quad {\color{#FF0000}{\cancel\text{EDGH}}} \quad {\color{#FD9000}{\cancel\text{EDHF}}} \quad {\color{#FF0000}{\cancel\text{EDHG}}}\\ {\color{#A800DD}{\cancel\text{EFDG}}} \quad {\color{#FD9000}{\cancel\text{EFDH}}} \quad {\color{#A800DD}{\cancel\text{EFGD}}} \quad {\color{#0000FF}{\text{EFGH}}} \quad {\color{#FD9000}{\cancel\text{EFHD}}} \quad {\color{#0000FF}{\cancel\text{EFHG}}}\\ {\color{#A800DD}{\cancel\text{EGDF}}} \quad {\color{#FF0000}{\cancel\text{EGDH}}} \quad {\color{#A800DD}{\cancel\text{EGFD}}} \quad {\color{#0000FF}{\cancel\text{EGFH}}} \quad {\color{#FF0000}{\cancel\text{EGHD}}} \quad {\color{#0000FF}{\cancel\text{EGHF}}}\\ {\color{#FD9000}{\cancel\text{EHDF}}} \quad {\color{#FF0000}{\cancel\text{EHDG}}} \quad {\color{#FD9000}{\cancel\text{EHFD}}} \quad {\color{#0000FF}{\cancel\text{EHFG}}} \quad {\color{#FF0000}{\cancel\text{EHGD}}} \quad {\color{#0000FF}{\cancel\text{EHGF}}}\\[0.8em] {\color{#A800DD}{\cancel\text{FDEG}}} \quad {\color{#FD9000}{\cancel\text{FDEH}}} \quad {\color{#A800DD}{\cancel\text{FDGE}}} \quad {\color{#009600}{\cancel\text{FDGH}}} \quad {\color{#FD9000}{\cancel\text{FDHE}}} \quad {\color{#009600}{\cancel\text{FDHG}}}\\ {\color{#A800DD}{\cancel\text{FEDG}}} \quad {\color{#FD9000}{\cancel\text{FEDH}}} \quad {\color{#A800DD}{\cancel\text{FEGD}}} \quad {\color{#0000FF}{\cancel\text{FEGH}}} \quad {\color{#FD9000}{\cancel\text{FEHD}}} \quad {\color{#0000FF}{\cancel\text{FEHG}}}\\ {\color{#A800DD}{\cancel\text{FGDE}}} \quad {\color{#009600}{\cancel\text{FGDH}}} \quad {\color{#A800DD}{\cancel\text{FGED}}} \quad {\color{#0000FF}{\cancel\text{FGEH}}} \quad {\color{#009600}{\cancel\text{FGHD}}} \quad {\color{#0000FF}{\cancel\text{FGHE}}}\\ {\color{#FD9000}{\cancel\text{FHDE}}} \quad {\color{#009600}{\cancel\text{FHDG}}} \quad {\color{#FD9000}{\cancel\text{FHED}}} \quad {\color{#0000FF}{\cancel\text{FHEG}}} \quad {\color{#009600}{\cancel\text{FHGD}}} \quad {\color{#0000FF}{\cancel\text{FHGE}}}\\[0.8em] {\color{#A800DD}{\cancel\text{GDEF}}} \quad {\color{#FF0000}{\cancel\text{GDEH}}} \quad {\color{#A800DD}{\cancel\text{GDFE}}} \quad {\color{#009600}{\cancel\text{GDFH}}} \quad {\color{#FF0000}{\cancel\text{GDHE}}} \quad {\color{#009600}{\cancel\text{GDHF}}}\\ {\color{#A800DD}{\cancel\text{GEDF}}} \quad {\color{#FF0000}{\cancel\text{GEDH}}} \quad {\color{#A800DD}{\cancel\text{GEFD}}} \quad {\color{#0000FF}{\cancel\text{GEFH}}} \quad {\color{#FF0000}{\cancel\text{GEHD}}} \quad {\color{#0000FF}{\cancel\text{GEHF}}}\\ {\color{#A800DD}{\cancel\text{GFDE}}} \quad {\color{#009600}{\cancel\text{GFDH}}} \quad {\color{#A800DD}{\cancel\text{GFED}}} \quad {\color{#0000FF}{\cancel\text{GFEH}}} \quad {\color{#009600}{\cancel\text{GFHD}}} \quad {\color{#0000FF}{\cancel\text{GFHE}}}\\ {\color{#FF0000}{\cancel\text{GHDE}}} \quad {\color{#009600}{\cancel\text{GHDF}}} \quad {\color{#FF0000}{\cancel\text{GHED}}} \quad {\color{#0000FF}{\cancel\text{GHEF}}} \quad {\color{#009600}{\cancel\text{GHFD}}} \quad {\color{#0000FF}{\cancel\text{GHFE}}}\\[0.8em] {\color{#FD9000}{\cancel\text{HDEF}}} \quad {\color{#FF0000}{\cancel\text{HDEG}}} \quad {\color{#FD9000}{\cancel\text{HDFE}}} \quad {\color{#009600}{\cancel\text{HDFG}}} \quad {\color{#FF0000}{\cancel\text{HDGE}}} \quad {\color{#009600}{\cancel\text{HDGF}}}\\ {\color{#FD9000}{\cancel\text{HEDF}}} \quad {\color{#FF0000}{\cancel\text{HEDG}}} \quad {\color{#FD9000}{\cancel\text{HEFD}}} \quad {\color{#0000FF}{\cancel\text{HEFG}}} \quad {\color{#FF0000}{\cancel\text{HEGD}}} \quad {\color{#0000FF}{\cancel\text{HEGF}}}\\ {\color{#FD9000}{\cancel\text{HFDE}}} \quad {\color{#009600}{\cancel\text{HFDG}}} \quad {\color{#FD9000}{\cancel\text{HFED}}} \quad {\color{#0000FF}{\cancel\text{HFEG}}} \quad {\color{#009600}{\cancel\text{HFGD}}} \quad {\color{#0000FF}{\cancel\text{HFGE}}}\\ {\color{#FF0000}{\cancel\text{HGDE}}} \quad {\color{#009600}{\cancel\text{HGDF}}} \quad {\color{#FF0000}{\cancel\text{HGED}}} \quad {\color{#0000FF}{\cancel\text{HGEF}}} \quad {\color{#009600}{\cancel\text{HGFD}}} \quad {\color{#0000FF}{\cancel\text{HGFE}}} \end{gathered} Let's finally list the possible combinations. Combinations DEFG DEFH DEGH DFGH EFGH We see that there are 5 possible combinations of 4 letters chosen from the list D, E, F, G, H.