Evaluate the given binomial coefficient. $\left(\frac{6}{6}\right)$

Verified step by step guidance

1. Identify the binomial coefficient notation given: \(\binom{6}{6}\), which represents the number of ways to choose 6 items from a set of 6 items.
2. Recall the formula for a binomial coefficient: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n!\) denotes the factorial of \(n\).
3. Substitute the values \(n = 6\) and \(r = 6\) into the formula: \(\binom{6}{6} = \frac{6!}{6!(6-6)!}\).
4. Simplify the expression inside the factorial in the denominator: \$6 - 6 = 0$, so the expression becomes \(\frac{6!}{6! \times 0!}\).
5. Recall that \$0!\( is defined as 1, so the expression simplifies to \(\frac{6!}{6! \times 1}\), which can be further simplified by canceling \)6!$ in numerator and denominator.

Key Concepts

Binomial Coefficient Definition

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 order. It is calculated using the formula C(n, k) = n! / (k! (n-k)!), where '!' denotes factorial.

Factorial Function

The factorial of a non-negative integer n, written as n!, is the product of all positive integers less than or equal to n. For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Factorials are essential in computing binomial coefficients.

Properties of Binomial Coefficients

One key property is that C(n, n) = 1 for any non-negative integer n, since there is exactly one way to choose all elements from a set. This property simplifies evaluation without full factorial calculation.