Evaluate the given binomial coefficient. (66)\binom66

Ch. 8 - Sequences, Induction, and Probability

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

모든 교과서

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 5

9장, 문제 5

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

검증된 단계별 안내

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.

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

Substitute the values $n = 6$ and $r = 6$ into the formula: $\binom{6}{6} = \frac{6!}{6!(6-6)!}$.

Simplify the expression inside the factorial in the denominator: $6 - 6 = 0$, so the expression becomes $\frac{6!}{6! \times 0!}$.

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Evaluate the given binomial coefficient. (66)\(\binom\)66