Binomial coefficient - practice problems

Combinatorial numbers, also known as binomial coefficients, represent the number of ways to choose k items from n items without regard to order. They are denoted as C(n,k), "n choose k," or using the notation (n k) in parentheses. These numbers appear in Pascal's triangle and are calculated using the formula n!/(k!(n-k)!). Combinatorial numbers are fundamental to probability theory, statistics, and the binomial theorem. They answer questions about selections, committees, and combinations in various contexts. Applications range from calculating lottery odds to analyzing genetic combinations and designing experiments.

Instructions: For each problem, solve carefully and show your complete working.