1. The problem is to understand the binomial coefficient $\binom{n}{k}$, which represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order. 2. The formula for the binomial coefficient is: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ (n factorial) is the product of all positive integers up to $n$. 3. Important rules: - $n$ and $k$ must be non-negative integers with $0 \leq k \leq n$. - $\binom{n}{0} = 1$ and $\binom{n}{n} = 1$. 4. To compute $\binom{n}{k}$, calculate the factorials of $n$, $k$, and $n-k$, then substitute into the formula. 5. Example: Calculate $\binom{5}{2}$. - Compute factorials: $5! = 120$, $2! = 2$, $3! = 6$. - Substitute: $\binom{5}{2} = \frac{120}{2 \times 6} = \frac{120}{12} = 10$. 6. This means there are 10 ways to choose 2 elements from 5. Final answer: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$