Evaluate the expression.

C(6, 4) · C(7, 5)
C(13, 9)

To evaluate the expression:

\[
\frac{C(6, 4) \cdot C(7, 5)}{C(13, 9)}
\]

We first calculate each combination separately using the formula for combinations:

\[
C(n, r) = \frac{n!}{r!(n-r)!}
\]

1. **Calculate \( C(6, 4) \):**

   \[
   C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{720}{24 \cdot 2} = \frac{720}{48} = 15
   \]

2. **Calculate \( C(7, 5) \):**

   \[
   C(7, 5) = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} = \frac{5040}{120 \cdot 2} = \frac{5040}{240} = 21
   \]

3. **Calculate \( C(13, 9) \):**

   \[
   C(13, 9) = \frac{13!}{9!(13-9)!} = \frac{13!}{9! \cdot 4!} = \frac{6227020800}{362880 \cdot 24} = \frac{6227020800}{8709120} = 715
   \]

Now, substitute the calculated values into the original expression:

\[
\frac{C(6, 4) \cdot C(7, 5)}{C(13, 9)} = \frac{15 \cdot 21}{715} = \frac{315}{715} = \frac{63}{143}
\]

The expression simplifies to \( \frac{63}{143} \).

Would you like more details on any part of the calculation or have any other questions?

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Here are 5 related questions:

1. How do you calculate the number of permutations for a given set?
2. What is the difference between combinations and permutations?
3. How would the expression change if we used \( C(6, 3) \) instead of \( C(6, 4) \)?
4. Can you express the original combination problem in a different context?
5. What is the binomial theorem, and how do combinations relate to it?

**Tip:** When solving combination problems, double-check the factorial calculations, especially for large numbers, to avoid mistakes.

Learn how to evaluate the expression C(6, 4) · C(7, 5) / C(13, 9) using combinations and factorials. This problem involves calculating specific combinations and understanding their application in mathematics. Suitable for high school students and those interested in advanced combinatorial mathematics.

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