The uploaded image shows a calculation involving permutations. Here's a detailed breakdown of the content:

Observations:

The expression being evaluated is the permutation formula: $P(n, r) = \frac{n!}{(n-r)!}$

This evaluates the number of permutations of $r$ objects chosen from a set of $n$ objects.

From the image:

1. First Calculation: $V^1_4 = \frac{4!}{(4-1)!} = \frac{24}{6} = 4$

2. Second Calculation: $V^2_4 = \frac{4!}{(4-2)!} = \frac{24}{2} = 12$

3. Third Calculation: $V^3_4 = \frac{4!}{(4-3)!} = \frac{24}{1} = 24$

4. Fourth Calculation: $V^4_4 = \frac{4!}{(4-4)!} = \frac{4!}{1} = 24$

Key Concept:

Here, $4! = 24$, which simplifies these calculations:

• $(4-1)! = 3! = 6$,
• $(4-2)! = 2! = 2$,
• $(4-3)! = 1! = 1$,
• $(4-4)! = 0! = 1$ (by definition).

The results align with the permutation rule.

Would you like me to explain how permutations work further or discuss a related topic?

5 Related Questions:

1. What is the difference between permutations and combinations?
2. How is the factorial function used in combinatorics?
3. Can you calculate $P(n, r)$ for larger values of $n$ and $r$?
4. What real-world scenarios can permutations be applied to?
5. How does the inclusion of repetition affect permutations?

Tip:

Always remember that in permutations, order matters, whereas in combinations, order does not matter.