Main Properties List:
In (x + y )n or ( 1 + x )n, we know the binomial coefficients are :
nC0,nC1, nC2, nC3,….. nCr… nCn.
In short, the binomial coefficients are also written as
C0, C1, C2, C3,….. Cr… Cn.
Let us now study a few salient properties of binomial coefficients:
Consider ( 1 + x )n = nc0 + nc1 . x + nc2 .x2 + nc3 . x3 +……..+ncn . xn………………(E)
To find the sum of the binomial coefficients nC0,nC1, nC2, nC3,….. nCr… nCn in (E) above, put x = 1.
- 2n = nc0 +nc1 + nc2 + …………. + ncn
- 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 C1, C3, C5 and so on are odd coefficients and C2, C4 , C6 and so on are even coefficients
- In (1 + x )n =nc0 + nc0 . x + nc2 . x2 + ……… + ncn . xn let us substitute -1 in x
- ( 1 – 1 )n = nc0 + nc1 . ( -1 ) + nc2 . (- 1 )2 + nc3 .(-1)3 + nc4 . (-1)4 + ……+ ncn .
(-1)n - 0 = nc0 – nc1 + nc2 – nc3 + nc4 +…………..+ ncn (-1)n
- Thus, nc0 + nc2 + nc4 +…………… = nc1 + nc3 + nc5 + ………….
- c0 + c2 + c4 +……….. = c1 + c3 + c5 + ………….
Because the sum of the both the odd and even binomial coefficients is equal to 2n, so the sum of the odd coefficients = ½ (2n ) = 2n – 1 , and sum of the even binomial coefficients = ½ (2n) = 2n – 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.
Number of terms in the following expansions:
- ( x + y )n is n + 1
- ( x + y + z ) n is [( n + 1 ) ( n + 2 )]/2
- ( x + y + z + w) n = [ ( n + 1)(n + 2 ) ( n + 3 )]/ 1. 2.3