Binomial Expansion Question Driving Me Mad

[demon8991]

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

 

[acm]

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.

 

[tacman]

.

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!