Binomial Expansion Question Driving Me Mad

In summary, the conversation is about a difficult question involving binomial expansion. The person has been struggling with it for hours and is looking for some help. The conversation then provides three solutions for the question: (1) using the binomial expansion formula, (2) using the observation that x^n-2 * x^2 = x^n, and (3) by equating coefficients. The third solution is the most efficient as it involves using the definition of (2n,n).
  • #1
demon8991
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Binomial Expansion Question Driving Me Mad!

http://img489.imageshack.us/img489/5239/binomialiv6.jpg

This is the last question on my maths sheet, and i must have been staring at it for hours, I've read all my notes and book but i just can't piece it together at all.

Driving me crazy!

You guys got any pointers, I am really confused
 
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  • #2
a) (1 + x)^n = (n,0) + (n,1)x + ... + (n,n)x^n
(1 + x)^2n = (1 + x)^n(1 + x)^n
Observe that x^n-2 * x^2 = x^n , the Hint shows that the coefficient (n,n-2) = (n,2) thus the coefficient of x^n will be (n,2)^2 and so forth for all powers of n. So it can be assumed that the coefficient of x^n = (n,0)^2 + (n,1)^2 + ... + (n,n)^2.

b) Obvious, by expansion. (1 + x)^2n = (2n, 0) + ... + (2n,n)x^n + ...+ (2n,2n)x^2n.

c)By equating coefficients (2n,n) = (n,0)^2 + (n,1)^2 + ... + (n,n)^2
The definition of (2n,n) = (2n)!/n!(2n-n)! = (2n)!/(n!)^2 as required.
 
  • #3
.

I completely understand how frustrating it can be when faced with a difficult math problem that just won't click. Here are a few pointers that may help you tackle this binomial expansion question:

1. Start by identifying the key components of the binomial expression. In this case, we have (2x + 3)^4, where 2x and 3 are the two terms in the binomial.

2. Remember the formula for binomial expansion: (a + b)^n = nC0 * a^n + nC1 * a^(n-1)b + nC2 * a^(n-2)b^2 + ... + nCn * b^n

3. Use the formula to expand (2x + 3)^4, keeping in mind that n = 4 and a = 2x, b = 3. This will give you a series of terms with coefficients.

4. Simplify each term by raising a to the appropriate power and combining like terms.

5. Don't forget to use the binomial coefficient (nCr) for each term. You can use a combination calculator or Pascal's triangle to find the coefficients quickly.

6. Once you have simplified each term, combine them to get your final answer.

Remember to take your time and double check your work. It's also helpful to write out each step and show your work, so you can catch any mistakes and understand the process better. Don't get discouraged, with a little perseverance and practice, you'll be able to tackle any binomial expansion question. Good luck!
 

Related to Binomial Expansion Question Driving Me Mad

1. What is binomial expansion?

Binomial expansion refers to the process of expanding a binomial expression, which is an algebraic expression with two terms. It involves raising each term to a certain power and then multiplying them together, resulting in a polynomial expression.

2. Why is binomial expansion important?

Binomial expansion is important in various fields such as mathematics, science, and statistics. It allows us to easily expand and simplify expressions, making it useful in solving equations and calculating probabilities.

3. How do you find the coefficients in binomial expansion?

The coefficients in binomial expansion can be found using Pascal's triangle or the binomial theorem. Pascal's triangle is a triangular arrangement of numbers that can be used to determine the coefficients, while the binomial theorem is a formula that can be used to calculate the coefficients.

4. What is the difference between binomial expansion and binomial distribution?

Binomial expansion is a mathematical process used to expand binomial expressions, while binomial distribution is a probability distribution that describes the likelihood of a certain number of successes in a fixed number of independent trials. They are related in that the coefficients in binomial expansion can be used to calculate probabilities in binomial distribution.

5. How can I use binomial expansion in real life?

Binomial expansion has various real-life applications such as in genetics, where it is used to calculate the probability of certain traits being inherited. It is also used in finance to calculate compound interest and in engineering to model the behavior of materials under different conditions.

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