we have 20 kinds of presents. we want to give 12 children presents but each child doesn't need to get one. No child can get 2 copies of the same present. in how many ways can we give presents?
please explain! the answer is (2^20)^12 but i don't see why
To determine the number of ways to give presents to 12 children from a set of 20 presents, we can break down the problem into multiple steps.
Step 1: Selecting presents for each child individually
Since every child can receive any number of presents (including none), we have 2 possibilities for each present for each child: either they receive it or they don't. This is similar to a binary choice, where each present corresponds to a bit and 0 represents the child not receiving the present, while 1 represents the child receiving the present.
Step 2: Combining the possibilities for all the children
To determine the total number of ways to give presents, we need to multiply the number of possibilities for each child together. Since there are 12 children, this means we multiply the number of possibilities for all the children together.
Step 3: Calculating the total number of possibilities
Each child has 2 possibilities for each present, and since there are 20 presents, this gives us 2^20 possibilities for each child. To calculate the total number of possibilities for all the children, we need to raise this value to the power of 12.
Therefore, the total number of ways to give presents is (2^20)^12.
Note that this assumes that the order in which the presents are given to the children does not matter. If the order is significant (e.g., Child A receiving present X and Child B receiving present Y is considered different from Child B receiving present X and Child A receiving present Y), then the total number of possibilities would be different.