In how many ways can 8 different dolls be packed in 5 identical gift boxes such that no box is empty if any of the boxes hold all of the toys?
Question
In how many ways can 8 different dolls be packed in 5 identical gift boxes such that no box is empty if any of the boxes hold all of the toys?
Solution
The problem can be solved using the concept of combinations in combinatorics.
Step 1: Since no box can be empty, we first place one doll in each box. We have 8 dolls, so we can do this in 8 choose 5 ways.
Step 2: Now we have 3 dolls left and 5 boxes. Each doll can go into any of the 5 boxes. So, for each doll, there are 5 choices of boxes. Since there are 3 dolls, the total number of ways is 5^3.
Step 3: Multiply the results from step 1 and step 2 to get the total number of ways the dolls can be packed in the boxes.
So, the answer is (8 choose 5) * 5^3.
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