In **(x + y ) ^{n} or ( 1 + x )^{n}**, we know the binomial coefficients are :

^{n}C_{0},^{n}C_{1}, ^{n}C_{2}, ^{n}C_{3},….. ^{n}C_{r}…
^{n}C_{n}.

In short, the binomial coefficients are also written as

** C _{0}, C_{1}, C_{2}, C_{3},….. C_{r}… C_{n}.**

Let us now study a few salient properties of binomial coefficients:

Consider

( 1 + x )^{n} = ^{n}c_{0} + ^{n}c_{1} . x + ^{n}c_{2} .x^{2} + ^{n}c_{3} . x^{3} +……..+^{n}cn . x^{n}………………(E)

To find the sum of the binomial coefficients ^{n}C_{0},^{n}C_{1}, ^{n}C_{2}, ^{n}C_{3},….. ^{n}C_{r}… ^{n}C_{n} in (E) above,

Put x = 1.

**1. 2 ^{n} = ^{n}c_{0} +^{n}c_{1} + ^{n}c_{2} + …………. + ^{n}c_{n}**

**2. Coefficients are odd or even based on the value of r.**

If the r values of terms are odd, then **binomial coefficients** of such terms are called odd coefficients and if the r values of terms are even then binomial coefficients of such terms are called even coefficient. Therefore

C_{1}, C_{3}, C_{5} and so on are odd coefficients and C_{2}, C_{4} , C_{6} and so on are even coefficients

In (1 + x )^{n} =^{n}c_{0} + ^{n}c_{0} . x + ^{n}c_{2} . x^{2} + ……… + ^{n}c_{n} . x^{n}

let us substitute -1 in x

( 1 – 1 )^{n} = ^{n}c_{0} + ^{n}c_{1} . ( -1 ) + ^{n}c_{2} . (- 1 )^{2} + ^{n}c_{3} .(-1)^{3} + ^{n}c_{4} . (-1)^{4} + ……+ ^{n}c_{n} .

(-1)^{n}

0 = ^{n}c_{0} - ^{n}c_{1} + ^{n}c_{2} – ^{n}c_{3} + ^{n}c_{4} +…………..+ ^{n}c_{n}
(-1)^{n}

Thus, ^{n}c_{0} + ^{n}c_{2} + ^{n}c_{4} +…………… = ^{n}c_{1} + ^{n}c_{3} + ^{n}c_{5} + ………….

**2. c _{0} + c_{2} + c_{4} +……….. = c_{1} + c_{3} + c_{5} + ………….**

Because the sum of the both the **odd and even binomial coefficients ** is equal to 2^{n}, so
the sum of the odd coefficients = ½ (2^{n} ) = 2^{n – 1} , and

Sum of the even binomial coefficients = ½ (2^{n}) = 2^{n – 1}

Thus, sum of the even coefficients is equal to the sum of odd coefficients.

In this way, we can derive several more properties of binomial coefficients by substituting suitable values for x and others in the binomial expansion.

**12. Number of terms in the following expansions:**

2. ( x + y + z )

3. ( x + y + z + w)