# Find Coefficient of x^9y^10 in (3x^3 - 4y^2)^8 | Combinatorics Problem Solution

• frankfjf
In summary, the conversation is about finding the coefficient of x^{9} y^{10} in (3x^{3} - 4y^{2})^{8}. The professor suggested a method of solving for each exponent, which the speaker attempted and obtained the answer C(8,5) * 3^{3} * (-4)^{5}. However, the answer key provided by the professor states that the answer is actually C(8,3) * 3^{3} * (-4)^{5}. After further examination, it is determined that both answers are equal and correct.

## Homework Statement

Find the coefficient of $$x^{9}$$ $$y^{10}$$ in (3$$x^{3}$$ - 4$$y^{2}$$)$$^{8}$$

## Homework Equations

The professor gave us a somewhat algebraic tactic or shortcut for solving these kinds of problems, mainly consisting of solving for each exponent. It can be somewhat tricky for me to explain with typing, but...

## The Attempt at a Solution

As per her suggested method, this is what I obtained:

C(8, k) (3$$x^{3}$$)$$^{8-k}$$ * ($$-4^{2}$$)$$^{k}$$

3(8 - k) = 9, 2k = 10. Thus k = 5 and 8 - k = 3.

This gives me the answer C(8,5) * $$3^{3}$$ * $$(-4)^{5}$$.

But her answer key in the provided review paper claims the answer is actually

C(8,3) * $$3^{3}$$ * $$(-4)^{5}$$

Have I done something wrong or is her answer incorrect? My text doesn't support her method and she is currently unavailable to assist me.

Last edited:
C(8,3)=C(8,5). If you swap k and 8-k you get the key answer. They are both equal and both right.

Checked it out, you are correct. Thank you for your assistance.

## 1. What is a coefficient?

A coefficient is a number that is multiplied by a variable in a mathematical expression. It represents the numerical value of the term containing the variable.

## 2. How do you find the coefficient of a specific term in a binomial raised to a power?

To find the coefficient of a specific term in a binomial raised to a power, you can use the binomial theorem or the combination formula. In this case, we can use the combination formula to find the coefficient of x^9y^10 in (3x^3 - 4y^2)^8.

## 3. What is the binomial theorem?

The binomial theorem is a formula that allows us to expand binomials raised to any power. It states that the coefficient of a term in the expansion is equal to the combination of the power and the term's exponents.

## 4. How do you use the combination formula to find the coefficient of a term in an expanded binomial?

The combination formula states that the coefficient of a term in an expanded binomial is equal to the combination of the power and the term's exponents. In this case, the power is 8 and the exponents of x^3 and y^2 are 9 and 10, respectively. Therefore, the coefficient of x^9y^10 is equal to 8 choose 9 (8C9) multiplied by (-4)^2 (since the coefficient of y^2 is -4). This simplifies to -4480.

## 5. Why is it important to find the coefficient of a term in a binomial expansion?

Finding the coefficient of a term in a binomial expansion allows us to determine the specific numerical value of that term. This is useful in various mathematical and scientific applications, such as calculating probabilities and solving equations.