Use the formula for nCr to evaluate each expression. ₅C₀

Verified step by step guidance

1. Recall the formula for combinations, which is given by \(nCr = \frac{n!}{r!(n-r)!}\), where \(n!\) denotes the factorial of \(n\).
2. Identify the values of \(n\) and \(r\) from the expression \$5C0\(, so here \)n = 5\( and \)r = 0$.
3. Substitute these values into the formula: \(5C0 = \frac{5!}{0!(5-0)!} = \frac{5!}{0! \cdot 5!}\).
4. Simplify the factorial expressions, remembering that \$0!$ is defined as 1, so the expression becomes \(\frac{5!}{1 \cdot 5!}\).
5. Cancel out the common factorial terms in the numerator and denominator to simplify the expression further.

Key Concepts

Combination Formula (nCr)

The combination formula, denoted as nCr, calculates the number of ways to choose r elements from a set of n elements without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is fundamental for counting problems in algebra and probability.

Factorial Function

The factorial of a non-negative integer n, written as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in calculating combinations and permutations, as they appear in the numerator and denominator of the nCr formula.

Evaluating Special Cases in Combinations

When evaluating combinations like nC0 or nCn, the result is always 1 because there is exactly one way to choose none or all elements from a set. Recognizing these special cases simplifies calculations and helps avoid unnecessary computation.