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In combinatorics, the combination formula is a key tool used to determine the number of ways to choose a subset of items from a larger set without considering the order of selection. This formula is written as: \( {}_{n} C_{r} = \frac{n!}{r!(n-r)!} \), where \(n\) stands for the total number of items and \(r\) represents the number of items to choose. The exclamation mark (!) denotes a factorial, which we will explain next. Using the combination formula, complicated counting problems become much simpler because it factors in all possible arrangements and then divides by the number of arrangements within each group.

A factorial calculation is essential for understanding combinations. The factorial of a number \(n\), represented as \(n!\), is the product of all positive integers from 1 to \(n\). For example, 4! (4 factorial) is calculated as:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Factorials grow very rapidly with increasing values of n, which is why they are so effective for calculations in combinatorics. When breaking down the combination formula, you'll often see factorials in both the numerator and the denominator. This can often simplify to a much smaller number due to common factors.

To complete an \( {}_{n} C_{r} \) calculation, follow these steps:
1. Identify \(n\) and \(r\): For \( {}_{30} C_{4} \), n=30 and r=4. 2. Substitute the values in the formula: \( {}_{30} C_{4} = \frac{30!}{4!(30-4)!} \). 3. Calculate the factorials: \(4! = 24\) and \(26!\) remains in the denominator to simplify. 4. Simplify the expression by canceling common factors: \( {}_{30} C_{4} = \frac{30 \times 29 \times 28 \times 27 \times 26!}{24 \times 26!} \). Notice that 26! cancels out. 5. Finally, perform the multiplication and division: \( \frac{30 \times 29 \times 28 \times 27}{24} = 27405 \).
This process shows how to break down a seemingly complex formula into a series of manageable steps.