Binomial Co-efficient

Binomial co-efficient comes from Binomial theorem in elementary algebra. Binomial co-efficient is immensely used to solve counting problems. If you already know about it you can't deny how useful it is for counting problems. Basically I am gonna present here some advanced properties of Binomial co-efficient. Before that it's good to know basics of Binomial theorem. Binomial is a Polynomial with only 2 terms. Polynomial with only 1 term called Monomial. Here are few examples:

Monomial: x2, 3x4

Binomial: x2 - y3, (x + y)6 , (x - y)3

Polynomial: 3x2 + y3 + xy

If you don't familiar with polynomial have a look at this link: Polynimial Basics

Let's get back to our context which is Binomial. What if we want to multiply a Binomial with itself several times ? That's implies we want to find out power of a Binomial. Suppose a binomial (x+y). We want to find (x+y)3. Binomial theorem helps us to do it easily and it exposes the result as a summation of multiple terms. Like this:

In above example notice co-efficients of the terms (1, 4, 6, 4, 1) and these co-efficients are known as Binomial co-efficient. It's time to see general formula of Binomial theorem:

Using summation notation it will look like:

When y = 1,

More specifically, when x = 2,

When x = 3,

Number of co-efficients is increased along with increasing power of binomial. Here is binomial co-efficient table:

       0    1    2    3    4    5    6    7
-------------------------------------------
0 |    0    0    0    0    0    0    0    0
1 |    1    1    0    0    0    0    0    0
2 |    1    2    1    0    0    0    0    0    
3 |    1    3    3    1    0    0    0    0
4 |    1    4    6    4    1    0    0    0
5 |    1    5    10   10   5    1    0    0
6 |    1    6    15   20   15   6    1    0
7 |    1    7    21   35   35   21   7    1

Now I am gonna present main content of this article which are properties/identities of binomial coefficient and these properties/identities are quite interesting to me.

Property 1: Symmetry Property

This property is very familiar to someone who knows binomial co-efficient.

Property 2: Pascal’s triangle Property

To calculate binomial co-efficient pascal triangle is very useful

Property 3: Monotonicity Property

Property 4: Sum of Binomial Co-efficient Property

Property 5:

Property 6:

Property 7: Alternating Sum Property

Property 8: Vendermonde Property

Property 9:

Property 10:

Property 11:

Property 12:

Please comment below if you find anything wrong in above article.

Blog Comments powered by Disqus.

Previous Post