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This expression represents the ratio of two combinations. The notation \( C(n, k) \) represents the combination or "binomial coefficient," defined as the number of ways to choose \( k \) items from \( n \) items without regard to order. It can be calculated with the formula:

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

In the given image, we have:

\[
\frac{C(42, 15)}{C(50, 15)} = \frac{\frac{42!}{(42 - 15)! \cdot 15!}}{\frac{50!}{(50 - 15)! \cdot 15!}}
\]

To simplify this ratio:

1. Substitute \( C(42, 15) \) and \( C(50, 15) \) with their factorial expressions:
   \[
   = \frac{\frac{42!}{27! \cdot 15!}}{\frac{50!}{35! \cdot 15!}}
   \]

2. The \( 15! \) terms in the numerator and denominator cancel out:
   \[
   = \frac{42! / 27!}{50! / 35!}
   \]

3. Simplify further by rewriting as a single fraction:
   \[
   = \frac{42! \cdot 35!}{50! \cdot 27!}
   \]

Calculating this directly would involve large numbers, so typically, the factorials can be simplified by expanding only the necessary terms or using computational tools to find the precise ratio.

Would you like to go into further details on simplification techniques or computational methods for evaluating such large factorials?

Here are some questions for further exploration:

1. What are some methods for simplifying large factorials in combinatorial expressions?
2. How can the gamma function generalize factorial calculations for non-integer values?
3. What practical applications do combinations have in probability and statistics?
4. How does this calculation change if we replace \( 15 \) with another value?
5. Why is the binomial coefficient significant in Pascal's triangle?

**Tip:** Use properties of factorials, like \( \frac{n!}{(n-k)!} = n \cdot (n-1) \cdot \ldots \cdot (n-k+1) \), to simplify expressions involving ratios of combinations.

Find the ratio of two combinations, C(42, 15) and C(50, 15).

Explore how to find the ratio of two combinations, C(42, 15) and C(50, 15), using factorials in combinatorics. This problem demonstrates techniques for simplifying factorial expressions, making it ideal for advanced high school students studying combinations and binomial coefficients.

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