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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.

Learn how to evaluate the permutation 7P4 step-by-step using the permutation formula nPr = n! / (n - r)!. This problem covers essential topics in permutations and factorials, suitable for students in Grades 9-12.

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