Evaluate the given binomial coefficient.$\left(\frac{8}{3}\right)$

Verified step by step guidance

1. Identify the binomial coefficient notation \( \binom{8}{3} \), which represents the number of ways to choose 3 elements from a set of 8 elements.
2. Recall the formula for the binomial coefficient: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \], where \( n! \) denotes the factorial of \( n \).
3. Substitute \( n = 8 \) and \( r = 3 \) into the formula: \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} \].
4. Express the factorials to simplify the expression: \[ 8! = 8 \times 7 \times 6 \times 5! \], so \[ \binom{8}{3} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} \].
5. Cancel the common \( 5! \) terms in numerator and denominator, then simplify the remaining expression \( \frac{8 \times 7 \times 6}{3!} \) to find the value.

Key Concepts

Binomial Coefficient

The binomial coefficient, denoted as (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 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. Factorials are essential in computing binomial coefficients and permutations, e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120.

Combinatorial Interpretation

Binomial coefficients count combinations, which are selections where order does not matter. Understanding this helps in problems involving probability, counting, and algebraic expansions like the binomial theorem.