Greatest Binomial Coefficient

Algebra > Binomial Theorem > Greatest Binomial Coefficient

9. Greatest Binomial Coefficients:

In each of the four binomial expansions below, the coefficients first increase and then start to decrease.

       1. ( x + y )2 = x2 + 2xy + y2
        2. ( x + y )3 = x3 + 3x2 y + 3xy2 + y3
        3. (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
        4. (x + y )5 = x5 + 5x4y + 10x3 y2 +10x2y3 +5xy3 + y5

For example, in ( x + y )5 above, the binomial coefficients start from 1, peak up to 10, and again fall to 1.

Let us write the values of all of the six binomial coefficients in( x + y )5 below:

5c0 = 1,
5c1 = 5,
5c2 = 10,
5c3 = 10,
5c4 = 5,
5c5 = 1

We note that the binomial coefficients always start with 1, rise to greatest value at the middle term or middle terms depending on whether index n is even or odd integer, and then again fall to 1 in the last term.

Now, let us learn a formula to find the greatest binomial coefficient:

Let the binomial expansion be in the form (1 + x ) n, where index n is a positive integer.
Two cases arise depending on whether the binomial index n is even or odd integer.

Case 1: When n is an odd integer:

Then there are two greatest binomial coefficients (these two are middle terms). They are:

nc(n+1)/2 ,nc( n + 3 )/2

Case 2: When n is an even integer.

Then there is only one greatest binomial coefficient (this is the only one middle term). It is: nc( n/2 + 1)

Eg 1. Find the greatest binomial coefficients in ( 1 + x )11

Solution:
since binomial index n is 11, an odd integer, therefore the greatest binomial coefficients are two and they are:
11c(11 + 1)/2 = 11c6
11c(11 + 3)/2 =11c7

Eg 2: Find the greatest binomial coefficient in the binomial expansion
( 1 + x )10

Solution:
Since index n is 10; there is only one greatest binomial coefficient . And it is

10c(10/2) + 1 = 10c6