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

Verified step by step guidance

1. Identify the binomial coefficient notation: \(\binom{12}{1}\), which represents the number of ways to choose 1 item from 12 items.
2. Recall the formula for a binomial coefficient: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n!\) denotes the factorial of \(n\).
3. Substitute \(n = 12\) and \(r = 1\) into the formula: \(\binom{12}{1} = \frac{12!}{1!(12-1)!}\).
4. Simplify the factorial expressions in the denominator: \$1! = 1\( and \)(12-1)! = 11!$, so the expression becomes \(\frac{12!}{1 \times 11!}\).
5. Recognize that \(\frac{12!}{11!} = 12\), so the binomial coefficient simplifies to \(12\).

Key Concepts

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.