In Exercises 1–8, evaluate the given binomial coefficient.

A binomial coefficient, denoted as (n choose k) or C(n, k), represents the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial, the product of all positive integers up to that number.

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, particularly in calculating permutations and combinations, including binomial coefficients.

The combinatorial interpretation of binomial coefficients provides a way to understand their significance in counting problems. For instance, (12 choose 1) represents the number of ways to select 1 item from a set of 12 distinct items, which is simply 12, as each item can be chosen individually. This interpretation is crucial for applying binomial coefficients in real-world scenarios.