The question asks for the number of permutations of 7 objects taken 4 at a time. This is a permutation problem where the order of selection matters.

The formula to calculate permutations is:

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

Where:

• $n$ is the total number of objects (7),
• $r$ is the number of objects taken at a time (4).

So, we have:

$P(7, 4) = \frac{7!}{(7 - 4)!} = \frac{7!}{3!}$

First, calculate the factorials:

• $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$,
• $3! = 3 \times 2 \times 1 = 6$.

Now, compute the permutation:

$P(7, 4) = \frac{5040}{6} = 840$

Thus, the number of permutations is 840.

Would you like further details or clarification on this? Here's a follow-up list of related questions:

1. How do permutations differ from combinations?
2. How do you calculate permutations when there are repeating items?
3. What is the importance of the order in permutation problems?
4. Can you solve a combination problem using a similar approach?
5. How would you approach a permutation problem with more than two factors?

Tip: Always remember that the key difference between permutations and combinations is that in permutations, order matters, while in combinations, order does not matter.