Browse · MATH
Printjmc
counting and probability senior
Problem
At the grocery store, I bought different items. I brought identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought in the identical bags, assuming he might leave some of the bags empty?
Solution
We can divide this into cases.
Case 1: All items go in the same bag. There is one possible way to do this.
Case 2: Three items go in one bag, and the last item goes in another bag. There are ways to choose which item goes in a bag by itself.
Case 3: Two items go in one bag, and the other two go in another bag. There are ways to choose which items go in the first bag, but since the bags are identical we must divide by to correct for overcounting. Therefore, there are arrangements in this case.
Case 4: Two items go in one bag, and the other two items each go in a different one of the remaining bags. There are ways to choose which two items are put in a bag together, and since the bags are identical it doesn't matter which bags the last two items are put in.
There are a total of different ways to put the items into bags.
Case 1: All items go in the same bag. There is one possible way to do this.
Case 2: Three items go in one bag, and the last item goes in another bag. There are ways to choose which item goes in a bag by itself.
Case 3: Two items go in one bag, and the other two go in another bag. There are ways to choose which items go in the first bag, but since the bags are identical we must divide by to correct for overcounting. Therefore, there are arrangements in this case.
Case 4: Two items go in one bag, and the other two items each go in a different one of the remaining bags. There are ways to choose which two items are put in a bag together, and since the bags are identical it doesn't matter which bags the last two items are put in.
There are a total of different ways to put the items into bags.
Final answer
14