Problem 41 - Algebra and Trigonometry

Question

A club with ten members is to choose three officers—president, vice president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?

Step-by-Step Solution

Answer

There are 720 ways to fill the three officer positions.

Step 1: Identify the number of elements to choose from

There are 10 members in the club. This will be our total number of elements that we can select from.

Step 2: Identify the number of positions to be filled

There are 3 positions to be filled: president, vice president, and secretary-treasurer. This will be the number of selections we need to make.

Step 3: Determine the type of combinatoric problem

Since the order of selection is important (that is distinguishing between president, vice president, and secretary-treasurer), this is a permutation problem.

Step 4: Apply the permutation formula

The formula for permutations is given by P(n, r) = n! / (n-r)!. Here, n is the total number of elements to choose from, and r is the number of selections to make. Therefore, we have P(10, 3) = 10! / (10-3)!.

Step 5: Compute the permutation

Computing this yields P(10, 3) = 10! / 7! = (10 × 9 × 8) = 720.

Key Concepts

Combinatorial Mathematics

Combinatorial mathematics is an area of mathematics that deals with counting, arrangement, and combination of sets of elements in a specific order. It has wide applications in various fields including computer science, cryptography, and optimization. One of the fundamental aspects of combinatorial mathematics is understanding how to calculate the number of possible arrangements or groupings of a set of items where the order of those items may or may not matter. These calculations often involve tools such as permutations, combinations, factorials, and binomial coefficients.

Considering our example of selecting officers for a club, combinatorial mathematics allows us to calculate exactly how many different ways there are to fill those roles considering the restrictions stipulated. In this scenario, it is key to recognize that the order of selection is significant; being chosen as president is not the same as being chosen as secretary-treasurer. Therefore, we use permutation calculations to find the solution as detailed in the exercise provided.

Permutation Formula

The permutation formula is critical in combinatorial mathematics when we want to know the number of ways to arrange 'r' objects from a larger set of 'n' objects where the order is important. It is represented as
\[ P(n, r) = \frac{n!}{(n-r)!} \]
where 'n' is the total number of objects to choose from, 'r' is the number to choose, 'n!' represents the factorial of 'n', and '(n-r)!' is the factorial of the difference between 'n' and 'r'.

For the club officer selection problem, we calculated the number of ways to choose 3 officers from 10 members using the permutation formula. Since the order is important (the roles of president, vice president, and secretary-treasurer are distinct), we look specifically at permutations rather than combinations, which would be used when order doesn't matter.

Factorial Notation

Factorial notation is essential in calculating permutations and combinations. It is denoted by an exclamation point (!) and indicates the product of all positive integers up to a given number. For example, \(5!\) (5 factorial) is \(5 \times 4 \times 3 \times 2 \times 1 = 120\). The factorial of zero, \(0!\), is defined to be 1.

In permutations, factorials determine the total number of ways a set can be ordered. The factorial notation simplifies the representation and calculation of permutations, such as in our example exercise where \(10!\) represents the total number of ways to order 10 people. When we calculate \(10! / 7!\) for the problem at hand, we are effectively cancelling out the factorial part for the officers not being chosen (7 of them) and thus focus on the factorial calculation for the 3 officers being selected.