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The problem requires calculating the value of \( M \) given by the formula:

\[
M = \frac{78!}{76! + 77!}
\]

Let's simplify this expression step by step.

1. **Simplify the factorial terms:**
   \[
   78! = 78 \times 77 \times 76!
   \]
   and
   \[
   77! = 77 \times 76!
   \]

2. **Rewrite the expression in terms of \(76!\):**
   \[
   76! + 77! = 76! + 77 \times 76! = (1 + 77) \times 76! = 78 \times 76!
   \]

3. **Substitute these into the formula for \(M\):**
   \[
   M = \frac{78 \times 77 \times 76!}{78 \times 76!}
   \]

4. **Simplify by canceling common terms:**
   \[
   M = \frac{78 \times 77}{78} = 77
   \]

Therefore, the correct answer is:

\[
\boxed{77 \text{ (Option c)}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify expressions involving factorials?
2. What are some common tricks to simplify large factorial expressions in problems?
3. How do you handle more complex factorial expressions involving sums and products?
4. Can you explain the concept of factorial growth and its implications in combinatorial problems?
5. How can you use approximations for large factorial calculations?

**Tip:** When dealing with factorials in expressions, look for common factors that can be canceled to simplify calculations!

Learn how to calculate the value of M using factorials with a step-by-step solution. Simplify the expression involving factorials and fractions to find the correct answer. Suitable for advanced math students.

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