Answer
If one item is selected from each of the four groups, number of ways one can order the meal = $4_{{C}{_{1}}}\times3_{{C}{_{1}}}\times4_{{C}{_{1}}}\times3_{{C}{_{1}}}$ = 144
Two orders are
In the first order we ordered Ham from main course, potatoes from vegetables, coffee from beverage and cake from dessert
In the second order we ordered chicken from main course, peas from vegetables, tea from beverage and pie from dessert
Work Step by Step
In the first order we ordered Ham from main course, potatoes from vegetables, coffee from beverage and cake from dessert
Let
The order from main course is Ham, then the number of ways Ham can be ordered (one out of 4) = $4_{{C}{_{1}}}$
The order from vegetables is Potatoes, then the number of ways Potatoes can be ordered ( one out of three) = $3_{{C}{_{1}}}$
The order from beverages is coffee, then the number of ways coffee can be ordered (one out of 4) = $4_{{C}{_{1}}}$
The order from desserts is cake, then the number of ways cake can be ordered ( one out of three) = $3_{{C}{_{1}}}$
If one item be selected from each of the four group, number of ways one can ordered the meal = $4_{{C}{_{1}}}\times3_{{C}{_{1}}}\times4_{{C}{_{1}}}\times3_{{C}{_{1}}}$ = 4 $\times$ 3 $\times$ 4 $\times$ 3 = 144
In order second we ordered chicken from main course, peas from vegetables, tea from beverage and pie from dessert
Let
The order from main course is chicken, then the number of ways chicken can be ordered (one out of 4) = $4_{{C}{_{1}}}$
The order from vegetables is Peas, then the number of ways Peas can be ordered ( one out of three) = $3_{{C}{_{1}}}$
The order from beverages is tea, then the number of ways tea can be ordered (one out of 4) = $4_{{C}{_{1}}}$
The order from desserts is pie, then the number of ways pie can be ordered ( one out of three) = $3_{{C}{_{1}}}$
If one item be selected from each of the four group, number of ways one can ordered the meal = $4_{{C}{_{1}}}\times3_{{C}{_{1}}}\times4_{{C}{_{1}}}\times3_{{C}{_{1}}}$ = 4 $\times$ 3 $\times$ 4 $\times$ 3 = 144
