The expression in the image is:

$4!^2 - C(15 - r)$

with the values provided as $r = 7$ and $r = 3$.

Let's break it down and evaluate each part.

1. Calculate $4!$:

   • $4! = 4 \times 3 \times 2 \times 1 = 24$.

2. Square $4!$:

   • $(4!)^2 = 24^2 = 576$.

So, the expression becomes:

$576 - C(15 - r)$

The $C(15 - r)$ term could indicate a combination or binomial coefficient, typically written as $C(n, k)$ or $\binom{n}{k}$, which represents the number of ways to choose $k$ items from $n$ items without regard to the order. Since only one variable is given, let’s consider the interpretation might involve calculating $C(15, r)$ where $r = 7$ or $r = 3$.

Case 1: $r = 7$

$576 - C(15, 7)$

Calculate $C(15, 7)$ using the formula for combinations:

$C(15, 7) = \frac{15!}{7!(15 - 7)!} = \frac{15!}{7! \times 8!}$

After calculating, $C(15, 7) = 6435$.

So for $r = 7$:

$576 - 6435 = -5859$

Case 2: $r = 3$

$576 - C(15, 3)$

Calculate $C(15, 3)$:

$C(15, 3) = \frac{15!}{3!(15 - 3)!} = \frac{15!}{3! \times 12!} = 455$

So for $r = 3$:

$576 - 455 = 121$

Summary of Results:

• For $r = 7$: the result is $-5859$.
• For $r = 3$: the result is $121$.

Would you like more detailed steps on any part of this calculation, or do you have any further questions?

Here are some related questions you might find helpful:

1. How do factorials work, and why is $4! = 24$?
2. What is the formula for combinations, and how is it applied?
3. Why does squaring $4!$ result in 576, and how is it relevant to the expression?
4. What are other applications of combinations in probability?
5. How can we evaluate other expressions with factorials and combinations?

Tip: Remember, combinations are used when order does not matter, while permutations are used when order does matter.