A Dance Instructor's Dilemma Unlocking Combinations For Student Selection

In the captivating world of dance, every performance is a symphony of movement, precision, and artistry. A skilled dance instructor plays a pivotal role in shaping this artistry, not only by imparting technical skills but also by curating the perfect ensemble of dancers for each performance. Imagine a scenario where a dance instructor has the delightful task of selecting four students from a pool of ten talented individuals to grace the stage. The question that arises is: In how many different ways can the instructor choose these four students if the order of selection doesn't matter?

This intriguing problem delves into the realm of combinatorics, a fascinating branch of mathematics that deals with counting and arranging objects. To unravel the solution, we'll embark on a journey through the fundamental concepts of combinations, permutations, and the powerful formula that governs them.

Understanding Combinations: The Art of Selection

At the heart of our problem lies the concept of combinations. Combinations are all about selecting a group of items from a larger set where the order of selection is irrelevant. In other words, whether we choose student A first and then student B, or student B first and then student A, the resulting group remains the same. This is precisely the scenario we face when the dance instructor selects students for the performance. The order in which they are chosen doesn't affect the final ensemble.

To grasp this concept more firmly, let's consider a simpler example. Suppose we have three students, Alice, Bob, and Carol, and we want to choose two of them for a duet. The possible combinations are:

• Alice and Bob
• Alice and Carol
• Bob and Carol

Notice that the order doesn't matter. Alice and Bob are the same duet as Bob and Alice.

Distinguishing Combinations from Permutations: Order Matters

Now, let's contrast combinations with another closely related concept: permutations. Permutations are arrangements of objects where the order of arrangement is crucial. Imagine that instead of selecting a group of students, the instructor is assigning specific roles in the performance, such as lead dancer, supporting dancer, and so on. In this case, the order of selection would matter because each position carries a unique significance.

To illustrate the difference, let's revisit our example of three students. If we were assigning the roles of lead dancer and supporting dancer, the possible permutations would be:

• Alice (lead), Bob (supporting)
• Alice (supporting), Bob (lead)
• Alice (lead), Carol (supporting)
• Alice (supporting), Carol (lead)
• Bob (lead), Carol (supporting)
• Bob (supporting), Carol (lead)

As you can see, there are more permutations than combinations because each order of arrangement is counted separately.

The Combination Formula: Unveiling the Calculation

Now that we've established the essence of combinations, let's delve into the formula that allows us to calculate them. The number of ways to choose k items from a set of n items (where order doesn't matter) is denoted by ${}_nC_k$ or inom{n}{k} and is calculated as follows:

{}_nC_k = rac{n!}{(n-k)!k!}

where "!" represents the factorial operation. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Let's break down the formula:

• n! (n factorial): This represents the total number of ways to arrange all n items in the set.
• (n-k)! ((n-k) factorial): This accounts for the number of ways to arrange the n-k items that are not chosen.
• k! (k factorial): This accounts for the number of ways to arrange the k items that are chosen. Since order doesn't matter in combinations, we divide by k! to eliminate duplicate arrangements.

Applying the Formula to the Dance Instructor's Dilemma

With the combination formula in our grasp, we can now tackle the dance instructor's challenge. We have n = 10 students, and the instructor needs to choose k = 4 of them. Plugging these values into the formula, we get:

{}_{10}C_4 = rac{10!}{(10-4)!4!} = rac{10!}{6!4!}

Let's calculate the factorials:

• 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
• 4! = 4 × 3 × 2 × 1 = 24

Now, substitute these values back into the equation:

{}_{10}C_4 = rac{3,628,800}{720 × 24} = rac{3,628,800}{17,280} = 210

Therefore, the dance instructor can choose four students from ten in 210 different ways.

Exploring the Significance of Combinations

The concept of combinations extends far beyond the realm of dance performances. It finds applications in various fields, including:

• Probability: Combinations are essential for calculating probabilities in situations where the order of events doesn't matter, such as drawing cards from a deck or selecting lottery numbers.
• Statistics: Combinations are used in sampling techniques to determine the number of possible samples that can be drawn from a population.
• Computer science: Combinations are employed in algorithms for generating subsets, solving optimization problems, and analyzing data.
• Genetics: Combinations play a role in understanding the inheritance of traits and the diversity of genetic combinations.

The Answer: A. 210

In conclusion, the dance instructor has 210 different ways to choose four students from ten for the performance. This problem beautifully illustrates the power of combinations in counting scenarios where order is irrelevant. By understanding the combination formula and its underlying principles, we can unlock solutions to a wide range of counting problems across diverse fields.

So, the next time you encounter a situation where you need to select a group of items without regard to order, remember the magic of combinations and the elegant formula that empowers us to count the possibilities.

Here are some of the keywords that this article used:

• Combinations
• Dance instructor
• Students
• Performance
• Order
• Factorial
• Permutations
• Probability
• Statistics
• Computer science
• Genetics