Problem 12 - Algebra and Trigonometry

Question

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{12} C_{5}\)

Step-by-Step Solution

Answer

The value of \(_{12}C_{5} = 792.

Step 1: Identify the values of n and r

Here, \(n = 12\) and \(r = 5\). We need to substitute these values into the combination formula.

Step 2: Evaluate factorials

First, calculate the factorial for \(n\), \(r\), and \(n-r\). So, we have \(12! = 479001600\), \(5! = 120\), and \((12-5)! = 7! = 5040\). Please note that factorial means multiplying that number by all positive integers less than the number, e.g. \(5! = 5*4*3*2*1 = 120\).

Step 3: Apply the combination formula

Substitute the computed values into the formula: \(_{12}C_{5} = \frac{12!}{5!(12-5)!} = \frac{479001600}{120*5040}\). Calculate the above expression to get the result.

Step 4: Perform the calculation and get the result

Performing the calculation gives \(_{12}C_{5} = 792.\)

Key Concepts

Factorials

A factorial is a fundamental concept in algebra, central to understanding permutations and combinations. Denoted by an exclamation point after a positive integer, the factorial of a number, say, for instance, \(n\), is the product of all positive integers from that number down to one. In other words, \(n! = n \times (n-1) \times (n-2) \times \text{...} \times 3 \times 2 \times 1\). It is important to remember that the factorial of zero is defined to be one, \(0! = 1\).

Factorials grow incredibly fast with increasing numbers. Therefore, it becomes difficult to compute them directly for larger numbers. However, they are essential when we need to find the number of ways in which items can be arranged or selected - a common problem in probability and statistics.

Combination Formula

The combination formula is used to find the number of ways in which you can choose \(r\) elements from a set of \(n\) elements, where order does not matter. The mathematical representation of the combination formula is \(_nC_r = \frac{n!}{r!(n-r)!}\). In this formula, \(n!\) represents the factorial of the total quantity of elements, while \(r!\) and \((n-r)!\) account for the number of chosen elements and the remaining elements, respectively.

Understanding the combination formula is crucial when dealing with problems where you need to pick a subset of items, like forming a committee from a group of people, without caring about the order of selection. It reveals the 'combinatorial power' of a set, allowing us to quantify possibilities in a structured way.

Permutations and Combinations

Permutations and combinations are both methods of counting arrangements within a set, but they differ in whether order is important. Permutations are concerned with the arrangement of objects where the order does matter, whereas combinations deal with the selection of objects where the order is irrelevant.

In permutations, the focus is on the arrangement of all or some members of a set in a specific sequence. The formula for permutations of \(r\) elements from a set of \(n\) elements is \(nP_r = \frac{n!}{(n-r)!}\). When it comes to combinations, we use the previously mentioned combination formula.

These two concepts are pivotal in probability, as they help calculate the likelihood of various events and in situations requiring sophisticated counting strategies, such as deciding the number of different passwords or seating arrangements.

Evaluating Expressions

Evaluating expressions, especially those involving factorials and combinations, requires methodical calculation. In the context of the given exercise, we evaluate expressions step by step - firstly identifying variables, then calculating factorials, and finally substituting these into the combinations formula. This process demands careful arithmetic calculation to avoid common errors, such as misunderstanding the factorial concept or misapplying the formula.

The ability to properly evaluate such expressions is a staple in algebra and is further used in higher levels of mathematics, physics, computer science, and many fields requiring data analysis. Mastery in this domain cultivates a strong foundation for solving complex problems related to counting and probability.