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

Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 1

9장, 문제 1

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

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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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주요 개념

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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.

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.

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