My question is "If I have 20 radio receivers, each with 3 channels to choose from, how many combinations of channels do I have (more than one receiver can have the same channel)", and I'm not sure whether the answer is 20^3, or 3^20 ... How do I know? Thanks!

Question

ybarrap · Answer

One  way to think about his is to do this mental exercise. Say you are one of these receivers and you have only 3 colors to choose from. So you have three choices, right? Obviously. Your partner also has three choices to choose from. Now, if you were looking at the ways you can combine your and your partner's choices in terms of COLOR, you would have your 3 choices and your partner's 3 choices, that would be $3 \times 3$ or 6 different ways to combine. Like red (R), yellow (Y), and blue (B). So your combinations would be RR,RY,RB,YY,YB,BB -- six choices. If there was another person, there would be $\large 3\times3\times3=3^3$, choices. And if there were 20 people, there would be $3^{20}$ choices. So the number of choices (colors) is always the base and the number of people (receivers) is always the exponent. 

The answer is  $3^{20}$.