Question

Carly has 15 stuffed animals. How many ways are there to organize her top 3 favorites?

Answer 1

Well, the answer will depend on whether the order will count or not (based on Permutations and Combinations). If the order counts, then we would use the formula for Permutations, which is:
$\frac{n!}{(n-r)!}$
Where n is the number of items you have, and r is the number of times you choose from the items.
$\frac{15!}{(15-3)!}$
Which simplifies to
$\frac{15!}{(12)!}$
Which simplifies to 15*14*13 (because all the numbers 1-12 in the factorial canceled out), which gets us the answer 2730.

Now, if you wanted to find the number of ways to order the toys without replacement (order doesn't count), you would use the formula:
$\frac{n!}{r!(n-r)!}$
The  variables are still the same, but you are now multiplying by r!.
$\frac{15!}{3!(15-3)!}$
Simplifies to
$\frac{15!}{3!(12)!}$
Which simplifies to (using the same cancellation method above)
$\frac{2730}{3!}$
Dividing 2730 by 3! will get us an answer of 455.

Really, it depends on whether they are ordered or not. In this case (since you didn't specify whether the order mattered), the answer would be 455 or 2730.

:)