Combinations And Derivatives Solving Math Problems Step By Step

Aug 5, 2025 by ADMIN 64 views

Unlocking Combinations and Derivatives A Deep Dive into Mathematical Problems

Hey guys! Ever find yourself staring blankly at a math problem, wondering where to even begin? Don't worry, we've all been there. Math can seem daunting, but breaking it down step by step can make even the trickiest questions manageable. In this article, we're going to tackle two intriguing problems: one involving combinations and the other delving into the world of derivatives. So, grab your thinking caps, and let's dive in!

Q1. Mastering Combinations: Computing ${ }^{15} C _3$

Let's start with our first challenge: computing combinations. The expression ${ }^{15} C _3$ might look intimidating at first, but it represents a fundamental concept in combinatorics – the art of counting! In essence, ${ }^{15} C _3$ asks: "How many ways can we choose a group of 3 items from a larger set of 15 items, where the order of selection doesn't matter?" Think of it like picking a team of 3 players from a squad of 15 – the order you pick them in doesn't change the team itself.

To solve this, we'll use the combinations formula, a powerful tool that mathematicians and problem-solvers have relied on for ages. This formula is given by:

${ }^{n} C _r = \frac{n!}{r!(n-r)!}$

Where:

n is the total number of items (in our case, 15)
r is the number of items we want to choose (in our case, 3)
! denotes the factorial, which means multiplying a number by all the positive integers less than it (e.g., 5! = 5 * 4 * 3 * 2 * 1)

Now, let's plug in our values into the formula:

${ }^{15} C _3 = \frac{15!}{3!(15-3)!} = \frac{15!}{3!12!}$

Time to unleash our factorial skills! We can expand the factorials as follows:

${ }^{15} C _3 = \frac{15 * 14 * 13 * 12 * 11 * ... * 1}{(3 * 2 * 1)(12 * 11 * ... * 1)}$

Notice something cool? We have 12! in both the numerator and the denominator, so they cancel each other out, simplifying our calculation significantly:

${ }^{15} C _3 = \frac{15 * 14 * 13}{3 * 2 * 1}$

Now, it's just a matter of arithmetic:

${ }^{15} C _3 = \frac{15 * 14 * 13}{6} = 5 * 7 * 13 = 455$

So, the answer is B. 455! There are 455 ways to choose a group of 3 items from a set of 15. Understanding combinations opens up a whole new world of possibilities, from calculating probabilities in games of chance to designing experiments in scientific research. It's a cornerstone of both theoretical and applied mathematics, and mastering it is a huge step towards becoming a math whiz.

Breaking Down the Combination Formula

Let's take a closer look at why the combination formula works. The numerator, $n!$ , represents the number of ways to arrange all n items in a specific order. However, since we're only interested in the groups themselves, and not the order within the group, we need to eliminate the arrangements that are essentially the same. That's where the denominator comes in.

$r!$ accounts for the number of ways to arrange the r items we've chosen. Since these arrangements don't change the group itself, we divide by $r!$ to avoid overcounting.
$(n-r)!$ accounts for the number of ways to arrange the remaining items that we didn't choose. These arrangements are also irrelevant to the group we've selected, so we divide by $(n-r)!$ as well.

By dividing by both $r!$ and $(n-r)!$ , we effectively eliminate all the redundant arrangements and arrive at the true number of unique combinations.

Real-World Applications of Combinations

Combinations aren't just abstract mathematical concepts; they have practical applications in various fields. Here are a few examples:

Probability: Combinations are used to calculate 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 statistical analysis to determine the number of possible samples that can be drawn from a population.
Computer Science: Combinations are used in algorithms for tasks like data mining and machine learning.
Game Theory: Combinations are used to analyze strategic games and determine the optimal moves.
Genetics: Combinations are used to calculate the probability of inheriting specific traits from parents.

As you can see, combinations are a powerful tool that can be applied to a wide range of problems. By understanding the concept of combinations and mastering the combination formula, you'll be well-equipped to tackle many real-world challenges.

Q2. Delving into Derivatives: Finding the Value of $\frac{d}{d x}(x^2)$ at M (1,0)

Alright, let's switch gears and dive into the exciting world of calculus, specifically derivatives. The derivative, denoted as $\frac{d}{dx}$ , represents the instantaneous rate of change of a function. In simpler terms, it tells us how much a function's output changes for a tiny change in its input. This is a powerful concept with applications ranging from physics and engineering to economics and finance.

In this problem, we're asked to find the value of the derivative of the function $f(x) = x^2$ at the point M(1,0). The notation $\frac{d}{dx}(x^2)$ means "the derivative of $x^2$ with respect to x".

To find the derivative, we'll use the power rule, a fundamental rule in calculus that states:

$\frac{d}{dx}(x^n) = nx^{n-1}$

Where n is any real number. This rule is your best friend when dealing with polynomial functions, and it's surprisingly easy to apply.

In our case, $n = 2$ , so applying the power rule gives us:

$\frac{d}{dx}(x^2) = 2x^{2-1} = 2x^1 = 2x$

Great! We've found the derivative of $x^2$ , which is $2x$ . But we're not done yet. We need to find the value of this derivative at the point M(1,0). Notice that the point M has coordinates (1,0), where 1 represents the x-coordinate and 0 represents the y-coordinate. Since we're evaluating the derivative with respect to x, we only need the x-coordinate.

So, we substitute $x = 1$ into our derivative:

$2x = 2(1) = 2$

Therefore, the value of $\frac{d}{d x}(x^2)$ at M(1,0) is 2. The answer is D. 2!

Understanding the Derivative as a Slope

But what does this value of 2 actually mean? The derivative at a point represents the slope of the tangent line to the function's graph at that point. Imagine drawing the graph of $f(x) = x^2$ , which is a parabola. At the point M(1,0), we can draw a line that touches the parabola at only that point – this is the tangent line. The slope of this tangent line is precisely the value of the derivative, which we found to be 2.

A slope of 2 means that for every 1 unit increase in x, the function's value (y) increases by 2 units. This gives us a visual understanding of how the function is changing at that specific point. A steeper slope indicates a faster rate of change, while a flatter slope indicates a slower rate of change.

Applications of Derivatives

Derivatives are not just abstract mathematical concepts; they have a wide range of practical applications in various fields. Here are a few examples:

Physics: Derivatives are used to calculate velocity (the rate of change of position) and acceleration (the rate of change of velocity).
Engineering: Derivatives are used in optimization problems, such as designing structures that can withstand maximum stress or minimizing the cost of production.
Economics: Derivatives are used to model economic growth, predict market trends, and analyze consumer behavior.
Finance: Derivatives are used to price financial instruments, manage risk, and optimize investment portfolios.
Computer Graphics: Derivatives are used to create realistic images and animations.

By understanding derivatives, you gain a powerful tool for analyzing and modeling real-world phenomena. From predicting the trajectory of a projectile to optimizing the design of a bridge, derivatives provide valuable insights into the rates of change that govern our world.

Wrapping Up

So, there you have it! We've successfully tackled two challenging math problems, one involving combinations and the other involving derivatives. We've seen how the combinations formula allows us to count the number of ways to choose groups of items, and how derivatives allow us to calculate instantaneous rates of change. These are just two examples of the many fascinating concepts in mathematics, and I hope this article has sparked your curiosity to explore more. Remember, math isn't about memorizing formulas; it's about understanding the underlying principles and applying them to solve problems. Keep practicing, keep exploring, and you'll be amazed at what you can achieve!