Question

Hayden wants to place 5 of his 8 trophies on the fireplace. How many ways can he arrange the trophies?

Answer 1

To find how many ways Hayden can arrange 5 out of 8 trophies, we can use the concept of permutations since the arrangement matters.

First, we need the formula for permutations, which is given by:

\[
P(n, r) = \frac{n!}{(n-r)!}
\]

Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to arrange. In this case, \( n = 8 \) and \( r = 5 \).

Plugging these values into the formula, we have:

\[
P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!}
\]

Now we will calculate \( 8! \) and \( 3! \):

\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\]

\[
3! = 3 \times 2 \times 1 = 6
\]

Now plug these values back into the permutation formula:

\[
P(8, 5) = \frac{40320}{6} = 6720
\]

Thus, the number of ways Hayden can arrange 5 of his 8 trophies on the fireplace is

\[
\boxed{6720}
\]