Exercise 23 Page 704

Now, because the order is not important in a combination, we will cross out any duplicate triplets. Keep in mind that any outcome with the same letters, regardless of the order, is the same triplet. For example, UVW and UWV are the same triplet. \begin{gathered} {\color{#0000FF}{\text{UVW}}} \quad {\color{#009600}{\text{UVX}}} \quad {\color{#FF0000}{\text{UVY}}} \quad {\color{#FD9000}{\text{UVZ}}}\\ {\color{#0000FF}{\cancel\text{UWV}}} \quad {\color{#A800DD}{\text{UWX}}} \quad {\color{#FF00FF}{\text{UWY}}} \quad \textcolor{#800000}{\text{UWZ}}\\ {\color{#009600}{\cancel\text{UXV}}} \quad {\color{#A800DD}{\cancel\text{UXW}}} \quad \textcolor{#00ff00}{\text{UXY}} \quad \textcolor{#000080}{\text{UXZ}}\\ {\color{#FF0000}{\cancel\text{UYV}}} \quad {\color{#FF00FF}{\cancel\text{UYW}}} \quad \textcolor{#00ff00}{\cancel\text{UYX}} \quad \textcolor{#7fffd4}{\text{UYZ}}\\ {\color{#FD9000}{\cancel\text{UZV}}} \quad \textcolor{#000080}{\cancel\text{UZX}} \quad \textcolor{#800000}{\cancel\text{UZW}} \quad \textcolor{#7fffd4}{\cancel\text{UZY}}\\[0.8em] {\color{#0000FF}{\cancel\text{VUW}}} \quad {\color{#009600}{\cancel\text{VUX}}} \quad {\color{#FF0000}{\cancel\text{VUY}}} \quad {\color{#FD9000}{\cancel\text{VUZ}}}\\ {\color{#0000FF}{\cancel\text{VWU}}} \quad \textcolor{#808000}{\text{VWX}} \quad \textcolor{#6495ed}{\text{VWY}} \quad \textcolor{#008b8b}{\text{VWZ}}\\ {\color{#009600}{\cancel\text{VXU}}} \quad \textcolor{#808000}{\cancel\text{VXW}} \quad \textcolor{#ff1493}{\text{VXY}} \quad \textcolor{#00bfff}{\text{VXZ}}\\ {\color{#FF0000}{\cancel\text{VYU}}} \quad \textcolor{#6495ed}{\cancel\text{VYW}} \quad \textcolor{#ff1493}{\cancel\text{VYX}} \quad \textcolor{#daa520}{\text{VYZ}}\\ {\color{#FD9000}{\cancel\text{VZU}}} \quad \textcolor{#008b8b}{\cancel\text{VZW}} \quad \textcolor{#00bfff}{\cancel\text{VZX}} \quad \textcolor{#daa520}{\cancel\text{VZY}}\\[0.8em] {\color{#0000FF}{\cancel\text{WUV}}} \quad {\color{#A800DD}{\cancel\text{WUX}}} \quad {\color{#FF00FF}{\cancel\text{WUY}}} \quad \textcolor{#800000}{\cancel\text{WUZ}}\\ {\color{#0000FF}{\cancel\text{WVU}}} \quad \textcolor{#808000}{\cancel\text{WVX}} \quad \textcolor{#6495ed}{\cancel\text{WVY}} \quad \textcolor{#008b8b}{\cancel\text{WVZ}}\\ {\color{#A800DD}{\cancel\text{WXU}}} \quad \textcolor{#808000}{\cancel\text{WXV}} \quad \textcolor{#adff2f}{\text{WXY}} \quad \textcolor{#ff4500}{\text{WXZ}}\\ {\color{#FF00FF}{\cancel\text{WYU}}} \quad \textcolor{#6495ed}{\cancel\text{WYV}} \quad \textcolor{#adff2f}{\cancel\text{WYX}} \quad \textcolor{#cd853f}{\text{WYZ}}\\ \textcolor{#800000}{\cancel\text{WZU}} \quad \textcolor{#008b8b}{\cancel\text{WZV}} \quad \textcolor{#ff4500}{\cancel\text{WZX}} \quad \textcolor{#cd853f}{\cancel\text{WZY}}\\[0.8em] {\color{#009600}{\cancel\text{XUV}}} \quad {\color{#A800DD}{\cancel\text{XUW}}} \quad \textcolor{#00ff00}{\cancel\text{XUY}} \quad \textcolor{#000080}{\cancel\text{XUZ}}\\ {\color{#009600}{\cancel\text{XVU}}} \quad \textcolor{#808000}{\cancel\text{XVW}} \quad \textcolor{#ff1493}{\cancel\text{XVY}} \quad \textcolor{#00bfff}{\cancel\text{XVZ}}\\ {\color{#A800DD}{\cancel\text{XWU}}} \quad \textcolor{#808000}{\cancel\text{XWV}} \quad \textcolor{#adff2f}{\cancel\text{XWY}} \quad \textcolor{#ff4500}{\cancel\text{XWZ}}\\ \textcolor{#00ff00}{\cancel\text{XYU}} \quad \textcolor{#ff1493}{\cancel\text{XYV}} \quad \textcolor{#adff2f}{\cancel\text{XYW}} \quad \textcolor{#ffd700}{\text{XYZ}}\\ \textcolor{#000080}{\cancel\text{XZU}} \quad \textcolor{#00bfff}{\cancel\text{XZV}} \quad \textcolor{#ff4500}{\cancel\text{XZW}} \quad \textcolor{#ffd700}{\cancel\text{XZY}}\\[0.8em] {\color{#FF0000}{\cancel\text{YUV}}} \quad {\color{#FF00FF}{\cancel\text{YUW}}} \quad \textcolor{#00ff00}{\cancel\text{YUX}} \quad \textcolor{#7fffd4}{\cancel\text{YUZ}}\\ {\color{#FF0000}{\cancel\text{YVU}}} \quad \textcolor{#6495ed}{\cancel\text{YVW}} \quad \textcolor{#ff1493}{\cancel\text{YVX}} \quad \textcolor{#daa520}{\cancel\text{YVZ}}\\ {\color{#FF00FF}{\cancel\text{YWU}}} \quad \textcolor{#6495ed}{\cancel\text{YWV}} \quad \textcolor{#adff2f}{\cancel\text{YWX}} \quad \textcolor{#cd853f}{\cancel\text{YWZ}}\\ \textcolor{#00ff00}{\cancel\text{YXU}} \quad \textcolor{#ff1493}{\cancel\text{YXV}} \quad \textcolor{#adff2f}{\cancel\text{YXW}} \quad \textcolor{#ffd700}{\text{YXZ}}\\ \textcolor{#7fffd4}{\cancel\text{YZU}} \quad \textcolor{#daa520}{\cancel\text{YZV}} \quad \textcolor{#cd853f}{\cancel\text{YZW}} \quad \textcolor{#ffd700}{\cancel\text{YZX}}\\[0.8em] {\color{#FD9000}{\cancel\text{ZUV}}} \quad \textcolor{#800000}{\cancel\text{ZUW}} \quad \textcolor{#000080}{\cancel\text{ZUX}} \quad \textcolor{#7fffd4}{\cancel\text{ZUY}}\\ {\color{#FD9000}{\cancel\text{ZVU}}} \quad \textcolor{#008b8b}{\cancel\text{ZVW}} \quad \textcolor{#00bfff}{\cancel\text{ZVX}} \quad \textcolor{#daa520}{\cancel\text{ZVY}}\\ \textcolor{#800000}{\cancel\text{ZWU}} \quad \textcolor{#008b8b}{\cancel\text{ZWV}} \quad \textcolor{#ff4500}{\cancel\text{ZWX}} \quad \textcolor{#cd853f}{\cancel\text{ZWY}}\\ \textcolor{#000080}{\cancel\text{ZXU}} \quad \textcolor{#00bfff}{\cancel\text{ZXV}} \quad \textcolor{#ff4500}{\cancel\text{ZXW}} \quad \textcolor{#ffd700}{\cancel\text{ZXY}}\\ \textcolor{#7fffd4}{\cancel\text{ZYU}} \quad \textcolor{#daa520}{\cancel\text{ZYV}} \quad \textcolor{#cd853f}{\cancel\text{ZYW}} \quad \textcolor{#ffd700}{\cancel\text{ZYX}} \end{gathered} Let's finally list the possible combinations. Combinations UVW UVX UVY UVZ UWX UWY UWZ UXY UXZ UYZ VWX VWY VWZ VXY VXZ VYZ WXY WXZ WYZ XYZ We see that there are 20 possible combinations of 3 letters chosen from the list U, V, W, X, Y, Z.