Evaluate the given binomial coefficient. (121)\binom{12}{1}

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 3

9장, 문제 3

Evaluate the given binomial coefficient. $(\frac{12}{1})$

검증된 단계별 안내

Identify the binomial coefficient notation: $\binom{12}{1}$, which represents the number of ways to choose 1 item from 12 items.

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

Substitute $n = 12$ and $r = 1$ into the formula: $\binom{12}{1} = \frac{12!}{1!(12-1)!}$.

Simplify the factorial expressions in the denominator: $1! = 1$ and $(12-1)! = 11!$, so the expression becomes $\frac{12!}{1 \times 11!}$.

Recognize that $\frac{12!}{11!} = 12$, so the binomial coefficient simplifies to $12$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

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. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in calculating permutations and combinations.

Properties of Binomial Coefficients

Binomial coefficients have properties such as (n choose 0) = 1 and (n choose 1) = n. These properties simplify calculations, for instance, (12 choose 1) equals 12, since choosing one element from twelve can be done in twelve ways.

Evaluate the given binomial coefficient. (121)\(\binom{12}{1}\)