Question

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:

\[
\boxed{32768}
\]