what are the uses of binomial theorem?

Answer 1

The binomial theorem is used to expand polynomials. So, if you're ever working with polynomials in any area of science, the binomial theorem will make things much easier for you.

More interestingly, understanding the binomial theorem is essential to understanding basic probability and combinatorics, two fields with very important real world applications. For example, if I flip a coin 5 times, the number of ways of getting 3 heads is the coefficient of x^3 in the expansion of:

(1+x)^5=(1+x)(1+x)(1+x)(1+x)(1+x)

This happens to be so because of the way algebra is done; just imagine that 1's represent tails and x's represent heads. Each time you flip you choose a 1 or an x. The coefficient of x^3 is known as Binomial(5,3)

More advanced cases of what I just demonstrated lead to generating functions, which are an extremely powerful technique for solving probability questions. For example, the probability of rolling a die 5 times and having the resulting rolls add up to 15 is the coefficient of x^15 in the expansion:

((x+x^2+x^3+x^4+x^5+x^6)/6)^5

You'd have a tough time solving this problem if you didn't know generating functions. The answer for this problem uses the same logic as the binomial theorem, but you have to understand why the binomial theorem works first.

Answer 2

The binomial theorem is very useful in calculating combinatorics and probability, as Rex said, above.
Here's a few easy links to get you started...
http://www.intmath.com/Series-binomial-theorem/Series-introduction.php
http://coweb.math.gatech.edu:8888/linear/710
Also it's useful in Taylor approximations and calculations regarding the theory of special relativity
http://mathworld.wolfram.com/BinomialCoefficient.html
(a bit technical, but at the bottom of this page is a huge list of links to various applications of the binomial theorem)
http://www.physicsforums.com/showthread.php?t=130074
Funny how the same questions keep popping up over and over. Just three weeks ago someone asked this same thing...
http://au.answers.yahoo.com/question/index?qid=20061201035728AALagdF

Answer 3

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Term One ======== 4C0 (2A)^4(B^2)^0 4!/(4-0)!0! 16A^4 (1) 16*A^4 Second Term ========== 4C1 (2A)^3(B^2)^1 4!/(4-1)!1! (2A)^3 (B^2)^1 4 * 8A^3 B^2 32 A^3B^2 Third Term ======== 4C2 (2A)^2 * (B^2)^2 4!/2!2! (2A)^2*(B^4) 6*4*A^2*B^4 24 A^2 B^4 Fourth Term ========= 4C3 (2A)^1 * (B^2)^3 4 * 2A * B^6 8AB^6 Fifth Term ======== 4C4 * (2A)^0 (B^2)^4 B^8 Comment ======= You should try and develop a pattern. 1. There are 5 terms, 1 more than the power. 2. The last powers for 2A and B^2 must add up to the power (in this case 4). 3. See if you can develop some pattern for 4Cx

Answer 4

binomial theorem helps you to expand a binomial to the power of any natural no.
It in turn helps us to fid the coefficient of any term

It is used in many places- in Physics, Calculus etc

Answer 5

I am in an algebra class right now and i can tell you in the real world binomials serve no propose

Answer 6

say u have a problem : expand (2x-5)^10

for you to do the normal way i.e.
(2x-5)*(2x-5)*(2x-5)*(2x-5)*...etc
takes too much valuable time.Hence the binomial theorem