Evaluate each expression.
$\frac{{}_4{C}_2{\cdot }_6{C}_1}{{}_{18}{C}_3}$

Verified step by step guidance

1. Identify the components of the expression: \(\frac{4C2 \cdot 6C1}{18C3}\). Here, \(nCr\) represents the combination formula, which calculates the number of ways to choose \(r\) items from \(n\) items without regard to order.
2. Recall the combination formula: \(nCr = \frac{n!}{r! (n-r)!}\), where \(n!\) denotes the factorial of \(n\).
3. Calculate each combination separately using the formula: - Calculate \(4C2 = \frac{4!}{2! (4-2)!}\) - Calculate \(6C1 = \frac{6!}{1! (6-1)!}\) - Calculate \(18C3 = \frac{18!}{3! (18-3)!}\)
4. Substitute the calculated values of \$4C2\(, \)6C1\(, and \)18C3$ back into the original expression \(\frac{4C2 \cdot 6C1}{18C3}\).
5. Simplify the resulting fraction by multiplying the numerator values and then dividing by the denominator value to get the final simplified expression.

Key Concepts

Combination Formula

The combination formula, denoted as nCr, calculates the number of ways to choose r items from n without regard to order. It is given by nCr = n! / [r!(n-r)!], where '!' denotes factorial. This concept is essential for evaluating expressions involving combinations.

Factorials

Factorials represent the product of all positive integers up to a given number n, written as n!. They are fundamental in calculating permutations and combinations, as they help determine the total number of arrangements or selections.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing expressions by factoring and canceling common terms. This skill is crucial when evaluating complex expressions like the given combination formula multiplied and divided by other terms.