Problem 43 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Choosing a Group In how many ways can six people be chosen from a group of ten?

Step-by-Step Solution

Answer

There are 210 ways to choose six people from ten.

Step 1: Understand the Concept of Combination

In combinatorics, a combination refers to the selection of items from a larger collection, where the order of selection does not matter. The formula to calculate combinations is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 2: Assign Values to Variables

For this problem, we need to select \( r = 6 \) people from a larger group of \( n = 10 \) people. We will substitute these values into the combination formula to find the number of ways to choose the group.

Step 3: Calculate Factorials

First, calculate the factorials required for the combination formula: - \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) - \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) - \( (10-6)! = 4! = 4 \times 3 \times 2 \times 1 \) .

Step 4: Substitute and Solve

Now that we have the factorials, use the combination formula: \[ C(10, 6) = \frac{10!}{6! \times 4!} \] Calculate \( 10! = 3628800 \), \( 6! = 720 \), and \( 4! = 24 \). Thus, \[ C(10, 6) = \frac{3628800}{720 \times 24} = \frac{3628800}{17280} = 210 \].

Step 5: Interpret the Result

The result of the combination calculation, 210, represents the number of different ways to choose 6 people from a group of 10, without regard to the order of selection.

Key Concepts

Factorial

The concept of a factorial is fundamental to understanding combinations. A factorial, denoted by an exclamation mark (!), is the product of an integer and all the integers below it. For example, a factorial of 5, written as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are crucial because they allow us to calculate how many different ways we can arrange a set of items. The formula for the factorial of any number \( n \) is expressed as \( n! \). This notation rapidly increases in value as \( n \) becomes larger. For instance, 10! equals 3,628,800. Each successive term involves multiplying by a larger base number, so the results grow quickly. Understanding factorial calculations is essential for solving problems in combinatorics, particularly those involving choosing groups of items from a larger set.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It provides tools and methods to solve various problems related to the structure and counting of sets. One of the primary applications of combinatorics is in calculating combinations, where the order of items does not matter, but the selection itself is key.

• In a combination, the number of ways to choose \( r \) items from \( n \) items is given by the formula: \( C(n, r) = \frac{n!}{r! \times (n-r)!} \).

This formula derives from the need to remove different permutations from consideration, since order is irrelevant in combinations. Each combination represents a unique set of elements selected from a larger pool, an operation common in probability, statistics, and in organizing data. By applying the concepts of combinatorics, we make informed predictions and decisions based on potential outcomes of grouped selections.

Choosing Groups

When faced with a task of choosing groups, it is essential to consider whether the order of selection matters. In the context of combinations, we are concerned with choosing elements without regard to order. This makes it different from permutations where order does play a role.
The given problem of selecting 6 people from a group of 10 is a classic example of a combination. Using the formula \( C(n, r) = \frac{n!}{r! \times (n-r)!} \), we assign \( n = 10 \) and \( r = 6 \) to find:

• \( C(10, 6) = \frac{10!}{6! \times (10-6)!} = \frac{10!}{6! \times 4!} \).

The calculated result, 210, tells us there are 210 possible ways to form a group of 6 people from 10. This solution showcases the elegance of combinations, which efficiently count the number of group selections possible in many practical and theoretical scenarios. Understanding this concept helps solve various problems in real-life applications, such as team selection, dividing tasks among workers, or organizing tournaments.