10. A dress-shop owner has 8 new dresses that she wants to display in the window. If the display window has 5 mannequins, in how many ways can she dress them up?

Answer 1

To determine the number of ways to dress up the 5 mannequins with the 8 new dresses, we'll consider that each mannequin can wear any of the 8 dresses. Since the dresses can be reused (the same dress can be used on multiple mannequins), we can use the counting principle for each mannequin.

Here's the step-by-step calculation:

1. Choices for the First Mannequin: There are 8 dresses available, so there are 8 choices.
2. Choices for the Second Mannequin: Again, there are 8 choices.
3. Choices for the Third Mannequin: Also 8 choices.
4. Choices for the Fourth Mannequin: Still, there are 8 choices.
5. Choices for the Fifth Mannequin: Finally, there are 8 choices here as well.

Since the choices for each mannequin are independent, we multiply the number of choices together:

\[ \text{Total Ways} = 8 \times 8 \times 8 \times 8 \times 8 = 8^5 \]

Now, we calculate \( 8^5 \):

\[ 8^5 = 32768 \]

Thus, the total number of ways the dress-shop owner can dress the 5 mannequins is:

\[ \boxed{32768} \]