Problem 6 - Algebra 2 and Trigonometry

Question

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects without replacement. \(r=2, n=12\)

Step-by-Step Solution

Answer

There are 132 ordered selections possible.

Step 1: Understand the Problem

The problem is asking for the number of ways to select 2 objects (in order) from a set of 12 objects. This is a permutation problem because the order of selection matters, and the selection is done without replacement.

Step 2: Apply the Permutation Formula

The formula for permutations of selecting \( r \) objects from \( n \) objects is given by: \( P(n, r) = \frac{n!}{(n-r)!} \). Here, \( r = 2 \) and \( n = 12 \).

Step 3: Calculate Factorials

Calculate the factorials needed: \( n! = 12! \) and \( (n-r)! = (12-2)! = 10! \). This simplifies the calculation to \( \frac{12!}{10!} \).

Step 4: Simplify the Expression

Since \( 12! = 12 \times 11 \times 10! \), we can simplify \( \frac{12!}{10!} \) to \( 12 \times 11 \) because the \( 10! \) terms cancel out.

Step 5: Final Calculation

Calculate \( 12 \times 11 = 132 \). Therefore, there are 132 ways to select 2 objects in order from 12.

Key Concepts

Understanding Factorials

Factorials are one of the foundational concepts in permutations and combinatorics. In essence, a factorial represents the product of all positive integers up to a certain number. It's denoted by the symbol "!". For instance, 5 factorial, written as 5!, means you multiply 5 by every positive integer less than itself: 5! = 5 × 4 × 3 × 2 × 1 = 120.

Factorials are important because they help in calculating the number of ways to arrange a set of objects. The factorial function grows rapidly and simplifies the process of creating ordered sets and permutations. In the context of permutations, factorials are used to calculate the total number of arrangements, ensuring every combination is counted in sequences where the order matters.

What are Ordered Selections?

Ordered selections are arrangements where the order of the chosen objects is important. This concept plays a critical role in permutation problems.

Consider a scenario where you have a lineup and the sequence you follow matters. For example, if you have three objects, A, B, and C, selecting two at a time can be done in these orders: AB, BA, AC, CA, BC, and CB. Each sequence is different due to the order, even though the same objects might be selected.

When solving a problem involving ordered selections without replacement, as in our exercise, listing out all possible sequences for smaller sets can be practical. However, for larger sets (like choosing 2 out of 12), it's more efficient to use permutation formulas. This approach helps easily calculate the number of such orderly arrangements.

Using the Permutation Formula

The permutation formula is fundamental when dealing with ordered selections. It calculates the number of possible sequences of a subset of objects from a larger pool, considering the importance of order.

The general formula is expressed as \( P(n, r) = \frac{n!}{(n-r)!} \), where:

• \(n\) is the total number of objects;
• \(r\) is the number of objects to select;
• \(n!\) is the factorial of the total objects;
• \((n-r)!\) is the factorial of the difference between total objects and selected objects.

In the exercise, we have \(n = 12\) and \(r = 2\). Plugging these into the formula gives \( P(12, 2) = \frac{12!}{10!} \). By cancelling out \(10!\) in the numerator and denominator, it simplifies to \(12 \times 11 = 132\). This result shows that there are 132 different ways to choose 2 ordered items from a collection of 12.