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

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 3

9장, 문제 3

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

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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 $12$.

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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. 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)\(\binom{12}{1}\)