1. The problem is to understand the meaning and properties of the binomial coefficient $\binom{n}{k}$. 2. The binomial coefficient $\binom{n}{k}$ represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order. 3. The formula for the binomial coefficient is: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$. 4. Important rules: - $\binom{n}{0} = 1$ for any $n$ because there is exactly one way to choose nothing. - $\binom{n}{n} = 1$ because there is exactly one way to choose all elements. - $\binom{n}{k} = 0$ if $k > n$ because you cannot choose more elements than are available. 5. Example: Calculate $\binom{5}{2}$: $$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = \frac{120}{12} = 10$$ This means there are 10 ways to choose 2 elements from 5. 6. In summary, $\binom{n}{k}$ is a fundamental concept in combinatorics used to count combinations.