https://file.aiquickdraw.com/mathbot/file/1730300324959p8qw59ho.jpg

The image contains an expression involving combinations (binomial coefficients). It reads:

\[
C_{13}^{4} + C_{5}^{3} = ?
\]

To solve this, let's evaluate each binomial coefficient separately:

1. **Calculate \( C_{13}^{4} \):**
   \[
   C_{13}^{4} = \frac{13!}{4!(13 - 4)!} = \frac{13!}{4! \cdot 9!}
   \]
   Simplifying, we only need the product of the first four terms in the numerator:
   \[
   C_{13}^{4} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = \frac{17160}{24} = 715
   \]

2. **Calculate \( C_{5}^{3} \):**
   \[
   C_{5}^{3} = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! \cdot 2!}
   \]
   Simplifying, we get:
   \[
   C_{5}^{3} = \frac{5 \times 4}{2 \times 1} = 10
   \]

3. **Add the results:**
   \[
   C_{13}^{4} + C_{5}^{3} = 715 + 10 = 725
   \]

### Final Answer:
\[
C_{13}^{4} + C_{5}^{3} = 725
\]

Would you like more details on binomial coefficients, or do you have any other questions?

### Related Questions:
1. What is the general formula for calculating binomial coefficients?
2. How do binomial coefficients relate to Pascal's Triangle?
3. Can you explain the application of binomial coefficients in probability?
4. What is the significance of combinations in real-life scenarios?
5. How do combinations differ from permutations?

### Tip:
Remember, the combination formula \( C_{n}^{k} \) represents the number of ways to choose \( k \) items from \( n \) without regard to order.

Learn how to solve the combination problem C(13, 4) + C(5, 3) with a detailed step-by-step solution. This problem covers essential concepts in combinatorics, including binomial coefficients and the combination formula. Suitable for high school students in Grades 9-12.

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