Re: A gift shop has 5 distinct display shelves and 6 ornaments to place on
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13 Aug 2026, 11:05
Because there are 6 objects and 5 shelves, and every shelf gets at least 1 object, exactly one shelf gets 2 objects, and the other 4 shelves get 1 object each.
First, pick which shelf gets the 2 objects:
5 choices for the shelf
Step 2: What can be on that 2 -object shelf?
There are only 3 possibilities for what goes on that special shelf:
2. Two Snow Globes (S, S)
1 way to pick the items.
Put the remaining 3 figurines on 3 open shelves ( \( 4 \times 3 \times 2=24\) ways).
The remaining snow globe fills the last shelf (1 way).
Total \(=1 \times 24=24\) ways.
One Snow Globe \& One Figurine (S, F)
3. ways to pick which figurine pairs with the snow globe.
Put the remaining 2 figurines on open shelves ( \(4 \times 3=12 \) ways).
The remaining 2 snow globes fill the last 2 shelves (1 way).
Total \(=3 \times 12=36 \) ways.
3. Two Figurines (F, F)
3 ways to pick which 2 figurines go together.
Put the 1 remaining figurine on an open shelf (4 ways).
The remaining 3 snow globes fill the last 3 shelves (1 way).
Total \(=3 \times 4=12\) ways.
Step 3: Add them up
Total ways to fill the objects \(=24+36+12=\mathbf{7 2}\) ways
Multiply by the 5 shelf options chosen in Step 1:
Total Arrangements \(=5 \times 72=\mathbf{3 6 0}\)