If ${ }^{3n} C_2 = 15$, Find The Value Of $n$.A. 5

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Introduction

In this article, we will delve into the world of binomial coefficients and explore a specific equation involving the combination of 3n choose 2. The equation is given as ${ }^{3n} C_2 = 15$, and our goal is to find the value of n that satisfies this equation. We will break down the solution into manageable steps, using mathematical concepts and formulas to arrive at the final answer.

Understanding Binomial Coefficients

Before we dive into the solution, let's take a moment to understand what binomial coefficients are. A binomial coefficient, often denoted as ${ }^{n} C_k$, represents the number of ways to choose k items from a set of n items without regard to order. It is calculated using the formula:

nCk=n!k!(n−k)!{ }^{n} C_k = \frac{n!}{k!(n-k)!}

where n! represents the factorial of n, which is the product of all positive integers up to n.

The Given Equation

The given equation is ${ }^{3n} C_2 = 15$. To solve for n, we need to manipulate the equation using the properties of binomial coefficients.

Step 1: Expand the Binomial Coefficient

Using the formula for binomial coefficients, we can expand ${ }^{3n} C_2$ as follows:

3nC2=(3n)!2!(3n−2)!{ }^{3n} C_2 = \frac{(3n)!}{2!(3n-2)!}

Step 2: Simplify the Expression

We can simplify the expression by canceling out common factors in the numerator and denominator:

3nC2=(3n)(3n−1)(3n−2)!2!(3n−2)!{ }^{3n} C_2 = \frac{(3n)(3n-1)(3n-2)!}{2!(3n-2)!}

Step 3: Cancel Out Common Factors

We can cancel out the (3n−2)!(3n-2)! terms in the numerator and denominator:

3nC2=(3n)(3n−1)2{ }^{3n} C_2 = \frac{(3n)(3n-1)}{2}

Step 4: Set Up the Equation

Now that we have simplified the expression, we can set up the equation:

(3n)(3n−1)2=15\frac{(3n)(3n-1)}{2} = 15

Step 5: Solve for n

To solve for n, we can start by multiplying both sides of the equation by 2:

(3n)(3n−1)=30(3n)(3n-1) = 30

Expanding the left-hand side of the equation, we get:

9n2−3n=309n^2 - 3n = 30

Rearranging the equation to form a quadratic equation, we get:

9n2−3n−30=09n^2 - 3n - 30 = 0

Step 6: Solve the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

n=−b±b2−4ac2an = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a = 9, b = -3, and c = -30.

Plugging in the values, we get:

n=−(−3)±(−3)2−4(9)(−30)2(9)n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(9)(-30)}}{2(9)}

Simplifying the expression, we get:

n=3±108918n = \frac{3 \pm \sqrt{1089}}{18}

n=3±3318n = \frac{3 \pm 33}{18}

Step 7: Find the Positive Solution

We are looking for a positive solution for n, so we will take the positive root:

n=3+3318n = \frac{3 + 33}{18}

n=3618n = \frac{36}{18}

n=2n = 2

Conclusion

In this article, we solved the binomial coefficient equation ${ }^{3n} C_2 = 15$ to find the value of n. We broke down the solution into manageable steps, using mathematical concepts and formulas to arrive at the final answer. The value of n that satisfies the equation is n = 2.

Final Answer

Q: What is a binomial coefficient?

A: A binomial coefficient, often denoted as ${ }^{n} C_k$, represents the number of ways to choose k items from a set of n items without regard to order. It is calculated using the formula:

nCk=n!k!(n−k)!{ }^{n} C_k = \frac{n!}{k!(n-k)!}

Q: How do I calculate the binomial coefficient?

A: To calculate the binomial coefficient, you can use the formula:

nCk=n!k!(n−k)!{ }^{n} C_k = \frac{n!}{k!(n-k)!}

where n! represents the factorial of n, which is the product of all positive integers up to n.

Q: What is the significance of the equation ${ }^{3n} C_2 = 15$?

A: The equation ${ }^{3n} C_2 = 15$ is a specific case of the binomial coefficient equation, where we are looking for the value of n that satisfies the equation. This equation has important applications in combinatorics and probability theory.

Q: How do I solve the equation ${ }^{3n} C_2 = 15$?

A: To solve the equation ${ }^{3n} C_2 = 15$, we can follow the steps outlined in the previous article:

  1. Expand the binomial coefficient using the formula.
  2. Simplify the expression by canceling out common factors.
  3. Set up the equation by equating the simplified expression to 15.
  4. Solve for n using the quadratic formula.

Q: What is the value of n that satisfies the equation ${ }^{3n} C_2 = 15$?

A: The value of n that satisfies the equation ${ }^{3n} C_2 = 15$ is n = 2.

Q: Can I use the binomial coefficient formula to solve other equations?

A: Yes, the binomial coefficient formula can be used to solve other equations involving combinations. The formula is a powerful tool for solving problems in combinatorics and probability theory.

Q: What are some real-world applications of binomial coefficients?

A: Binomial coefficients have many real-world applications, including:

  • Combinatorial problems, such as counting the number of ways to arrange objects.
  • Probability theory, such as calculating the probability of certain events.
  • Statistics, such as analyzing data and making predictions.
  • Computer science, such as designing algorithms and data structures.

Q: Can I use a calculator to solve binomial coefficient equations?

A: Yes, you can use a calculator to solve binomial coefficient equations. Many calculators have built-in functions for calculating combinations and permutations.

Q: What are some common mistakes to avoid when working with binomial coefficients?

A: Some common mistakes to avoid when working with binomial coefficients include:

  • Forgetting to cancel out common factors.
  • Not using the correct formula for the binomial coefficient.
  • Not checking the validity of the solution.

Conclusion

In this article, we answered some frequently asked questions about binomial coefficients and the equation ${ }^{3n} C_2 = 15$. We covered topics such as calculating binomial coefficients, solving the equation, and real-world applications of binomial coefficients. We also provided some tips and common mistakes to avoid when working with binomial coefficients.