Fox, Goose, Bag of Beans: Why 16 Possibilities?

Question

I don't understand why there are $2^4$ possibilities in this problem and not $4!$ possibilities. Why is permutation not the correct approach in this case?

Answer 1

If you need to know the number of orders you can put $4$ things in, that's $4!$. On the other hand, if you're counting the number of ways that $4$ things can each be in one of two states, that will be $2^4$.

These are both instances of the basic counting principle: The number of ways a sequence of events can happen is the product of the numbers of ways each can happen.

In an ordering problem, the four events would be: Place an item first. Place an item second. Place an item third. Place an item fourth. The number of options available at each stage leads us to the product: $4\times 3\times 2\times 1$.

If you want to know how many ways four things can be in one of two states (such as how many outcomes for four coin flips, or how many ways to place 4 characters on two river banks), the four events that occur are: Place the farmer on one shore or the other. Place the fox on one shore or the other. Place the goose on one shore or the other. Place the beans on one shore or the other. The number of options available at each stage leads us to the product: $2\times 2\times 2\times 2$.

Does that help?