Question

A Sunday school class of 12 members Is to be seated on seven chairs and a bench that accommodates five persons. In how many ways can the bench be occupied?

Short answer

The bench can be occupied in 792 different ways. This is determined using the combinations formula \(C(n, r) = \frac{n!}{r!(n-r)!}\), where n is the total number of options (in this case, 12 members), r is the number to be chosen (5 for the bench). After calculating the factorials and substituting them into the formula, we find that \(C(12, 5) = 792\).

Step-by-step solution

Set up the combination formula

The combination formula is given by: \(C(n, r) = \frac{n!}{r!(n-r)!}\), where "n!" represents the factorial of n (i.e., the product of all positive integers up to n). In our problem, we need to find the number of ways to choose 5 people from a group of 12, so the combination we are looking for is: \(C(12, 5)\).

Calculate the factorials

Now we will calculate the factorials of n, r, and (n-r) to plug into our formula. n! = 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 r! = 5! = 5 × 4 × 3 × 2 × 1 (n-r)! = (12-5)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Apply the combination formula

Using the calculated factorials, we can now apply the combination formula: \(C(12, 5) = \frac{12!}{5!(7!)}\).

Calculate the result

Now we just need to substitute the factorials into the formula and calculate the result: \(C(12, 5) = \frac{12!}{5!(7!)} = \frac{12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{(5 × 4 × 3 × 2 × 1)(7 × 6 × 5 × 4 × 3 × 2 × 1)}\) After canceling out common factors, we get: \(C(12, 5) = \frac{12 × 11 × 10 × 9 × 8}{5 × 4 × 3 × 2 × 1} = \frac{12 × 11 × 10 × 9 × 8}{120} = 792\) So, there are 792 different ways to choose 5 people from a group of 12 members for seating on the bench.

Key concepts

Combination Formula

Understanding the combination formula is key in solving problems related to selecting items or individuals from a group where the order does not matter. This mathematical principle is symbolized as \(C(n, r) = \frac{n!}{r!(n-r)!}\), where \(n\) represents the total number of items, and \(r\) is the number of items being chosen. The exclamation mark (!) denotes factorial, another essential math concept we will discuss later.

For instance, if a teacher wants to pick a team of 3 students out of a class of 10, they would use the combination formula to find out how many distinct groups could be formed: \(C(10, 3)\). The formula ensures that selecting student A, B, and C in any order will still count as one single group. This principle is incredibly useful when planning events, creating teams, and in many other real-world scenarios.

Factorial Calculation

In mathematics, a factorial is the product of an integer and all the positive integers below it and is denoted by the symbol (!). For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials are not just a mathematical curiosity; they are instrumental in calculating permutations and combinations, especially when you need to figure out the number of different ways items can be arranged or selected. The factorial of 0 is defined as 1, and this definition helps to simplify many mathematical expressions. Understanding factorials is an important foundation for anyone looking to master the art of counting and arranging objects in mathematics.

Permutations and Combinations

Permutations and combinations are two fundamental mathematical concepts used in counting and probability. Permutations focus on arrangements where the order is important – think of lining up books on a shelf or assigning starting positions in a race.

Understanding Permutations

In permutations, every detail about the arrangement of the objects matters. The formula for calculating permutations is \(P(n,r) = \frac{n!}{(n-r)!}\).

Understanding Combinations

Combinations, on the other hand, deal with group selection where the order does not matter. As we've seen with the combination formula, it is used whenever the arrangement is irrelevant. The practical applications of these two concepts are widespread, seen in everything from sports team drafts to cryptographic algorithms.

Mathematical Counting Methods

Mathematical counting methods encompass various techniques from fundamental principles to more specific counting formulas like permutations and combinations. These methods allow us to systematically count the number of ways certain events can occur without having to list every single possibility.

For example, consider a simple scenario where you have a pair of dice. To determine the number of possible outcomes, you would use fundamental counting principles, multiplying the number of sides on one die by the number on the other (6 sides each die, so \(6 \times 6 = 36\) outcomes). When the scenarios become more complex, such as dealing with larger groups or specific conditions, combinations and permutations provide a streamlined method of calculation. Learning to select the appropriate counting method is crucial for tackling various real-life and theoretical problems in mathematics.