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. This leaves us with 3 dolls.
Step 2: Now, we need to distribute these 3 dolls in 5 boxes. This is a problem of distributing n identical items (dolls) into r different groups (boxes), which can be solved using the formula for combinations with repetition: C(n+r-1, r-1).
Step 3: Substituting n=3 (remaining dolls) and r=5 (boxes) into the formula, we get C(3+5-1, 5-1) = C(7, 4).
Step 4: Calculate the value of C(7, 4). This is equal to 7! / [4!(7-4)!] = 35.
So, there are 35 ways to pack 8 different dolls in 5 identical gift boxes such that no box is empty if any of the boxes hold all of the toys.
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