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

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.
  • #1
168
0

Homework Statement



Find the coefficient of [tex]x^{9}[/tex] [tex]y^{10}[/tex] in (3[tex]x^{3}[/tex] - 4[tex]y^{2}[/tex])[tex]^{8}[/tex]

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[tex]x^{3}[/tex])[tex]^{8-k}[/tex] * ([tex]-4^{2}[/tex])[tex]^{k}[/tex]

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

This gives me the answer C(8,5) * [tex]3^{3}[/tex] * [tex](-4)^{5}[/tex].

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

C(8,3) * [tex]3^{3}[/tex] * [tex](-4)^{5}[/tex]

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:
Physics news on Phys.org
  • #2
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.
 
  • #3
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.

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

Back
Top