Question regarding Binomial Theorem

In summary, The conversation discusses a question about finding the coefficient of x^n in the expansion of (1+x)^n(1+x)^n. The attempt at a solution involves using the binomial expansion and considering the series of squared coefficients, but the relation between the two is unclear. The question asker is looking for guidance and suggests using induction.
  • #1
mess1n
24
0
Hey people, I've racked my brain on this question for hours and can't seem to get to grips with it, wondering if i could get a little guidance?

Homework Statement



Considering the co-efficient of x^n in the expansion of (1+x)^n(1+x)^n, show:

[the sum from k=0 -> k=n of:] [nCk]2 = 2nCn

The Attempt at a Solution



i've found the co-efficient of x^n by using the binomial expansion on (1+x)^2n, and i can see that the binomial expansion of (1+x)^2n will give a series of squared co-efficients, but I have no idea how to relate the two :s.

Any help would be much appreciated.

Cheers,
Andrew
 
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  • #2
Can't you use induction?
 

Related to Question regarding Binomial Theorem

1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical theorem that describes the expansion of a binomial expression raised to a positive integer power. It is commonly used to simplify and solve problems involving binomial expressions.

2. What is the formula for the Binomial Theorem?

The formula for the Binomial Theorem is (a + b)^n = a^n + nC1 * a^(n-1) * b + nC2 * a^(n-2) * b^2 + ... + nCn * b^n, where n is a positive integer and nCr represents the combination formula nCr = n! / (r!(n-r)!).

3. How is the Binomial Theorem used in real life?

The Binomial Theorem has many real-life applications, such as in probability and statistics, finance, and physics. It is used to calculate the probability of certain outcomes in a given scenario, to model the growth of investments or populations, and to analyze physical phenomena such as projectile motion.

4. Can the Binomial Theorem be extended to negative or fractional powers?

Yes, the Binomial Theorem can be extended to negative or fractional powers using the generalized binomial theorem formula (a + b)^r = a^r + rC1 * a^(r-1) * b + rC2 * a^(r-2) * b^2 + ... + rCn * b^r, where r is any real number.

5. What are some common misconceptions about the Binomial Theorem?

One common misconception about the Binomial Theorem is that it only applies to two-term expressions (binomials). However, it can also be used for multi-term expressions (polynomials). Another misconception is that it can only be used for positive integer powers, when in fact it can be extended to any real number exponent. Additionally, some people may think that the Binomial Theorem is only useful for theoretical purposes, but it has many practical applications in various fields.

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