In each of the 4 binomial expansions below, the coefficients first increase and then start to decrease.
- ( x + y )2 = x2 + 2xy + y2
- ( x + y )3 = x3 + 3×2 y + 3xy2 + y3
- (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
- (x + y )5 = x5 + 5x4y + 10×3 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