**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} = x^{2} + 2xy + y^{2}**

2. ( x + y )^{3} = x^{3} + 3x^{2} y + 3xy^{2} + y^{3}

3. (x + y)^{4} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}

4. (x + y )^{5} = x^{5} + 5x^{4}y + 10x^{3} y^{2} +10x^{2}y^{3} +5xy^{3} + y^{5}

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:

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:

^{n}c_{(n+1)/2} ,^{n}c_{( 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:
^{n}c_{( 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:

^{11}c_{(11 + 1)/2} = ^{11}c_{6}

^{11}c_{(11 + 3)/2} =^{11}c_{7}

**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

^{10}c_{(10/2) + 1} = ^{10}c_{6}