When do i use the power ^ sign in a combination question?

Question

My problem: If there are 5 different candies in a jar and a child wants to take out one or more candies, how many ways can this be done?

I said it is $^5C_1 -\; ^5C_0 = 5-1 = 4$ ways. The $-1$ for the unwanted case using this trick:

At least/At most = total number of combinations - unwanted cases

But according to my answer sheet, it said $2^5 -1$ is the answer.

So my question is that in what situations should I use exponents and what impact does it have?

Answer 1

Generally you use a power whenever you have several independent choices to make, and each choice has the same options.

Here, for each candy in the jar you can choose to take it, or not ($2$ options). All these decisions are independent, and there are $5$ of them, so there are $2^5$ possibilities for which candies you can take from the jar. However, this includes the case of taking none of them, so you need to subtract $1$ to get the answer you're looking for.

Answer 2

Adding to the other answers:

$^5C_1-\, ^5C_0$ doesn't make sense for the following reasons:

• The problem states that the child takes one or more candies. None of these quantities in the expression includes data about the case where more than one candy is taken.

• $^5C_1$ gives the number of ways to select one candy. On the other hand, $^5C_0$ is the number of ways to select $0$ candies. It doesn't make sense to subtract these two because they correspond to different events.

Answer 3

Think of it like this: The child can either take a specific candy or not take it. This means we have $2$ possibilities for whether this candy is taken or not. Given we have $5$ candies, we have $2\cdot2\cdot2\cdot2\cdot2=2^5$ ways of taking $5$ candies. Since the condition that we have $0$ candies can be ignored, we have $2^5-1$ candies. Using $^5C_1-^5C_0$ would work if we could pick only one candy. (although the $^5C_0$ is unnecessary)