Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Classify each problem according to whether it involves a permutation or a combination. A four-member executive committee is to be formed from a twelve-member board of directors. In how many ways can it be formed?

Step-by-Step Solution

Answer

This is a combination problem, as the order of the members doesn't matter. Using the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), with n = 12 and k = 4, we can calculate that there are 495 ways to form a four-member executive committee from a twelve-member board of directors.

Step 1: Determine if it is a Permutation or Combination

In this problem, we are selecting a committee, so the order of the members doesn't matter. This means that we are dealing with a combination problem, as we are concerned with selecting a group of items without regard to the order.

Step 2: Apply the Combination Formula

The combination formula is given by the following expression: \[C(n, k) = \frac{n!}{k!(n-k)!}\] where n is the total number of items, k is the number of items we want to choose, and C(n, k) represents the number of ways to choose k items from a set of n items. In our case, there are 12 members on the board of directors (n = 12), and we need to form a committee of 4 members (k = 4).

Step 3: Calculate the Number of Ways

By plugging the values into the combination formula, we can calculate the number of ways to form the committee. \[C(12, 4) = \frac{12!}{4!(12-4)!}\] \[C(12, 4) = \frac{12!}{4!8!}\] Now, let's calculate the factorials: \(12! = 12\times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) \(4! = 4 \times 3 \times 2 \times 1\) \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) Plugging the factorials into the combination formula, we get: \[C(12, 4) = \frac{12\times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1)(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}\] Notice that some terms cancel out in the numerator and denominator: \[C(12, 4) = \frac{12\times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1}\] Finally, perform the arithmetic: \[C(12, 4) = \frac{11880}{24} = 495\] So, there are 495 ways to form a four-member executive committee from a twelve-member board of directors.

Key Concepts

Combination Formula

Understanding the combination formula is essential when determining the number of ways in which a certain group can be chosen from a larger set. A combination does not take into account the order in which items are selected, making it distinct from permutations where the order does matter.

The combination formula is represented by:
\[C(n, k) = \frac{n!}{k!(n-k)!}\]
Here, \(n\) indicates the total number of items to choose from, and \(k\) represents the number of items you want to select. The term \(n!\) is the factorial of \(n\), which is the product of all positive integers up to \(n\). The symbol \(C(n, k)\) is also known as a binomial coefficient, showcasing its roots in binomial theorem applications.

For instance, when we want to form a committee and the arrangement of members doesn't matter, we use this combination formula. Suppose we have 12 board members and need to choose a committee of 4, we use the formula to find the number of unique groups of four that can be made, which in this case is 495.

Factorial Calculation

Factorial calculation is a fundamental operation in combinatorics that helps in solving many problems involving permutations and combinations. The notation \(n!\) represents a factorial, which is the product of all positive integers up to \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

This operation becomes valuable when reducing the combination formula where factorials in the numerator and denominator often cancel out. It's essential to remember that zero factorial, \(0!\), is defined as 1. When using the combination formula, as seen in our earlier example with the 12-member board, factorial calculation simplifies the process significantly by allowing cancelation of terms which makes the formula easier to compute.

Combinatorics in Mathematics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It plays a vital role in various areas such as number theory, algebra, and geometry.

One key aspect of combinatorics is understanding how objects relate to each other when grouped or ordered in specific ways. While the combination formula relates to the grouping of objects where order isn't important, permutations would be used when the order does matter. These concepts are used widely in probability theory, statistics, and computer science for algorithm design and analysis.

To enhance practical understanding of these concepts, it's important to solve a variety of problems, this not only deepens comprehension but also builds problem-solving skills. For those struggling to understand a concept, breaking it down into smaller parts and applying it to real-world scenarios can often make the subject matter more relatable and easier to digest.