If 4 people have five meal choices how many different ways can they order? If 4 people have five meal choices, and multiple people can have the same meal. How many different ways can they order?

There are $(5^4)$ possible ways in which four people may order their meals, assuming everyone orders one meal. (Remember, everyone may order the same meal.) 
Person 1: 5 choices 
Person 2: 5 choices 
Person 3: 5 choices 
Person 4: 5 choices 
Multiply: $5 \cdot 5\cdot 5\cdot 5 = 5^4 = 625$ possible meal orders for 4 people.
If some can opt to fast (not order anything), so everyone orders at most one meal, then there are $(6^4)$ possible ways four people may order their meals, since now each of the 4 people actually have $6$ choices, one of which is simply not ordering anything offered.
