Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

jahnavi

  • 2 years ago

what is binomial theorem?

  • This Question is Closed
  1. Pr0bability
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It's a theorem that comes from combinatorics. Binomial is a polynomial with two (bi) terms. It basically says this: \[\left( a + b \right)^{n}=\sum_{m=0}^{n} \left( \left(\begin{matrix} n \\ m \end{matrix}\right) a ^{n-m} b ^{m}\right)\] So for example: \[\left( a + b \right)^{3}=\left(\begin{matrix}3 \\ 0\end{matrix}\right) a ^{3} b ^{0} + \left(\begin{matrix}3 \\ 1\end{matrix}\right) a ^{2} b ^{1} + \left(\begin{matrix}3 \\ 2\end{matrix}\right) a ^{1} b ^{2} +\left(\begin{matrix}3 \\ 3\end{matrix}\right) a ^{0} b ^{3} \] Which is nothing else then: \[a ^{3}+3a ^{2}b+3ab ^{2}+b ^{3}\] It can be proven with mathematical induction.

  2. jahnavi
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am not able to understand,can explain in a more simpler way?

  3. MattBenjamins
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    A binomial is (as explained above) an expression that has two terms. The general form is: \[(a+b)^{n}\] Where a is the first term, b the second and n is the power to which the whole gets raised. If n is low expanding the expression is easily done. For example: \[(x+2)^2 = (x+2)(x+2) = x^2+2x+2x+4 = x^2+4x+4\] When n is high though, this kind of expansion is a lot of work and there is an easier way to do it which is called the Binomial Theorem: \[(a+b)^n = \sum_{m=0}^{n}\left(\begin{matrix}n \\ m\end{matrix}\right)a^{n-1}b^m\] The first "scary bit" is what's called the Sigma Notation. In it's geneal form: \[\sum_{k=1}^{n}a _{k}\] This is just a simpler way to write the summation of many terms. For example: \[\sum_{k=1}^{4}2k=2+4+6+8=20\] In this example there are four terms in which some operation is aplied to k (in this case multiplication by two) and in the first term k=1. For each following term 1 gets added to k. The second thing is: \[\left(\begin{matrix}n \\ m\end{matrix}\right)\] Read: 'n choose m'. This means: the number of m sized sets you can take out of an n number of objects when order doesn't matter. Don't worry if that sounds confusing, to use the binomial theorem you only need to be able to calculate it which is done as follows: \[\left(\begin{matrix}n \\m\end{matrix}\right)=\frac{ n! }{ m!(n-m)! }\] Where n! (read n factorial) means:\[1\times2\times3\times...\times(n-2)\times(n-1)\times(n )\]For example: \[7! = 1\times2\times3\times4\times5\times6\times7\times=5040\] So for example: \[\left(\begin{matrix}4 \\ 2\end{matrix}\right)=\frac{ 4! }{ (2!)\times(2!) }=\frac{ 24 }{4 }=6\] Note that when m=0: \[\left(\begin{matrix}n \\0\end{matrix}\right)=1\] Hopefully this explanation can help you understand the example in the firts reply.

  4. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.