Evaluate the given binomial coefficient.(83)\binom{8}{3}

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 1

Capitolo 9, Problema 1

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

Guida verificata passo dopo passo

Identify the binomial coefficient notation $ \binom{8}{3} $, which represents the number of ways to choose 3 elements from a set of 8 elements.

Recall the formula for the binomial coefficient: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \], where $ n! $ denotes the factorial of $ n $.

Substitute $ n = 8 $ and $ r = 3 $ into the formula: \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} \].

Express the factorials to simplify the expression: \[ 8! = 8 \times 7 \times 6 \times 5! \], so \[ \binom{8}{3} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} \].

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.

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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.

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Special Products - Cube Formulas

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.

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Factorials

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.

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Fundamental Counting Principle

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