The image asks to evaluate the expression $^7P_4$.

This notation represents a permutation, specifically the number of ways to arrange 4 items selected from a set of 7 distinct items. The formula for permutations is:

$^nP_r = \frac{n!}{(n - r)!}$

For $^7P_4$:

$^7P_4 = \frac{7!}{(7 - 4)!} = \frac{7!}{3!}$

First, calculate $7!$ and $3!$:

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

Now, divide:

$^7P_4 = \frac{5040}{6} = 840$

So, the solution is $840$.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do permutations differ from combinations?
2. How is the permutation formula derived?
3. Can you explain how $^nP_0$ is defined and calculated?
4. What happens to the permutation value when $r = n$?
5. How do you calculate permutations for larger values of $n$?

Tip: In permutation problems, order matters. For combinations, order does not.