Question

Evaluate the expression. \({ }_{12} C_3\)

Short answer

The expression \({ }_{12} C_3\) evaluates to 220.

Step-by-step solution

Define the values

In this exercise, we have n=12 and r=3. This means we are choosing 3 items from a total set of 12.

Apply the combination formula

Using the formula \(\frac{n!}{(n-r)!*r!}\), we can substitute the values we defined earlier: \(\frac{12!}{(12-3)!*3!}\).

Simplify factorials

First, calculate the factorial values. The factorial of a number is the product of every integer from 1 to that number. So, 12! = 479001600, (12-3)! = 9! = 362880, and 3! = 6. Substitute these values into the formula to get \(\frac{479001600}{362880*6}\).

Perform division

Performing the division gives the final result: 220.

Key concepts

Factorials

The concept of factorials is fundamental to understanding combinatorics. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. For instance, the factorial of 5, written as 5!, is calculated by multiplying 5 × 4 × 3 × 2 × 1, which equals 120.
Factorials grow extremely fast, particularly for larger numbers. By the time you reach just 10!, it equals 3,628,800.
In combinatorics, factorials are utilized in various formulas to calculate permutations and combinations. They are used to determine the total number of ways items can be arranged or chosen, which is a core aspect of solving these mathematical problems.

Combination Formula

The combination formula is a key tool in combinatorics to calculate the number of ways a specified number of items can be chosen from a larger set, where the order of selection does not matter. The formula for a combination is written as:

• C(n, r) = \(\frac{n!}{r!(n-r)!}\)

This formula requires substituting the factorials of the total number of items \(n\) and the number of items to choose \(r\). For example, to solve \({ }_{12} C_3\), you'd calculate \(\frac{12!}{3! \cdot (12-3)!}\).
The combination formula is especially useful for problems where you're grouping or selecting items without regard to their arrangement, such as selecting team members or collecting groups of objects.

Combinatorial Analysis

Combinatorial analysis is the branch of mathematics dealing with counting, arranging, and selecting objects. It's utilized to solve complex problems by breaking them down into more manageable steps using specific rules and formulas.
Some common areas where combinatorial analysis is applied include:

• Calculating probabilities
• Solving puzzles and games
• Designing networks

In the context of our example problem, combinatorial analysis helped break down the steps to evaluate the combination \({ }_{12} C_3\), allowing us to find the total number of ways to choose 3 items from a set of 12 using a systematic approach.
This discipline not only aids mathematicians but also computer scientists and engineers in developing algorithms and optimizations.