To determine the number of possible permutations of 8 objects taken 4 at a time, we use the permutation formula [tex]\(P(n, r)\)[/tex].
The permutation formula is given by:
[tex]\[
P(n, r) = \frac{n!}{(n-r)!}
\][/tex]
where [tex]\(n\)[/tex] is the total number of objects, and [tex]\(r\)[/tex] is the number of objects taken at a time.
Here, [tex]\(n = 8\)[/tex] and [tex]\(r = 4\)[/tex].
1. First, calculate [tex]\(n!\)[/tex] which is [tex]\(8!\)[/tex]:
[tex]\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\][/tex]
2. Then, calculate [tex]\((n-r)!\)[/tex] which is [tex]\((8-4)!\)[/tex]:
[tex]\[
(8-4)! = 4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
3. Now, apply the permutation formula:
[tex]\[
P(8, 4) = \frac{8!}{(8-4)!} = \frac{40320}{24} = 1680
\][/tex]
Therefore, the number of possible permutations of 8 objects taken 4 at a time is 1,680.
The correct answer is:
○ C. 1,680
