Problem 77 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A quorum (minimum) of 6 voting members is required at all meetings of the Curtis Townhomes Owners Association. If there is a total of 12 voting memhers in the group, find the numher of ways this quorum can be formed.

Step-by-Step Solution

Answer

There are 924 different ways to form a quorum of 6 voting members from a total of 12 members in the Curtis Townhomes Owners Association.

Step 1: Identify n and k

n represents the total number of voting members, which is 12. k represents the size of the quorum, which is 6. Therefore, n = 12 and k = 6.

Step 2: Calculate factorials

Now, we need to calculate the factorial of n (n!), the factorial of k (k!), and the factorial of (n-k) ((n-k)!). \(n! = 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600\) \(k! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\) Since n = 12 and k = 6, (n-k) = (12-6) = 6. \((n-k)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

Step 3: Calculate the number of combinations

Now that we have the values for n!, k!, and (n-k)!, we can plug them into the combinations formula: \[C(n,k) = \frac{n!}{k!(n-k)!} = \frac{479001600}{720 \times 720} = \frac{479001600}{518400} = 924\] So, there are 924 different ways to form a quorum of 6 voting members from a total of 12 members in the Curtis Townhomes Owners Association.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, often denoted by the exclamation mark (!) after a number. The factorial of a positive integer is the product of all positive integers less than or equal to it. For instance:

\(4! = 4 \times 3 \times 2 \times 1 = 24\).

Factorials are especially important in the field of combinatorics, where they play a critical role in calculating permutations and combinations. The concept may seem straightforward, but it’s vital not to forget that the factorial of zero, \(0!\), is defined as 1, which often comes in handy during calculations. In the aforementioned exercise involving the Curtis Townhomes Owners Association, factorials are used to determine how many ways the group’s quorum can be achieved. Comprehending the concept of factorials is key to solving many problems in applied mathematics and helps provide the building blocks for more advanced mathematical concepts and operations.

Permutations and Combinations

Permutations and combinations are ways to count the number of different possibilities for arranging and selecting items. While both deal with sets of items, they serve different purposes:

Permutations

These are arrangements of items where the order does matter. For example, the sequence 'ABC' is different from 'BCA'. The general formula for permutations of a set of n items taken r at a time is given by \(P(n,r) = \frac{n!}{(n-r)!}\).

Combinations

These are selections of items where the order doesn't matter. The example from the Curtis Townhomes Owners Association, where the concern is about the group of 6 members and not the order in which they are chosen, is an instance of a combination. The formula for combinations is \(C(n,k) = \frac{n!}{k!(n-k)!}\), which allows us to find how many groups of k items can be chosen from a larger set of n items.

Understanding the distinction between permutations and combinations is essential for solving a variety of problems in applied mathematics, and it helps students in recognizing when the order of selection is a factor in the problem at hand.

Applied Mathematics

Applied mathematics involves the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. This field, therefore, is crucial for practical problem-solving in a wide array of professional areas. Factorials, permutations, and combinations, the core concepts discussed above, are tools of applied mathematics used for analyzing and predicting outcomes and making decisions based on the possible arrangements of data or objects. The exercise regarding the voting members of the Curtis Townhomes Owners Association is an example of how applied mathematics comes into play in everyday decision-making processes. It’s about translating real-world problems into mathematical language, solving them with mathematical methods, and interpreting the results back in real-world terms. This discipline requires not only an understanding of the theoretical concepts but also the capacity to translate theory into practice.