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

Verified step by step guidance

1. Identify the binomial coefficient formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
2. Substitute the given values into the formula: \( n = 6 \) and \( k = 6 \)
3. Calculate the factorials: \( 6! \) and \( (6-6)! = 0! \)
4. Recall that \( 0! = 1 \)
5. Simplify the expression: \( \binom{6}{6} = \frac{6!}{6! \cdot 1} \)

Key Concepts

Binomial Coefficient

A binomial coefficient, denoted as C(n, k) or 'n choose 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.

Factorial

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.

Properties of Binomial Coefficients

Binomial coefficients have several important properties, such as C(n, 0) = 1 for any n, and C(n, n) = 1, which indicates that there is exactly one way to choose all or none of the elements. Additionally, they satisfy the symmetry property C(n, k) = C(n, n-k), reflecting the idea that choosing k elements is equivalent to leaving out n-k elements.