1.Coefficient of x^m in the expansion of {ax^p + b/x^q}ⁿ is the coefficient of T_r+1 where   r=np-m/p+q

 

to get the independent term of the above expansion put m=0

 

2.The coefficient of x^n-1 in the expansion of (x-1)(x-2)(x-3).....(x-n) is    -n(n+1)/2.

 

3.The coefficient of x^n-1 in the expansion of (x+1)(x+2)....(x+n) is    n(n+1)/2.

 

4.If the coefficients of  r^th,(r+1)^th,(r+2)^th terms of (1+x)ⁿ are in A.P. then

   (n-2r)²=n+2

 

5.The r^th term from the end of (1+x)ⁿ is the (n-r+2)^th term from the beginning

 

6.Sum of the coefficients of (ax+by)ⁿ is (a+b)^n.

 

7. The no.of rational terms in the expansion of (^{[p ]}a+^{[q ]}b)ⁿ is

 [ n/L.C.M of p&q] +1 where[.] =step function.

 

8.The no.of non-zero terms in the expansionj of

 

i} (x+a)ⁿ + (x-a)ⁿ is       1) n+1/2   if n is odd

 

                                  2)n/2+1 if n is even

 

ii}(x+a)ⁿ - (x-a)ⁿ is       1)n+1/2  if n is odd

 

                                 2)n/2  if n is even

 

9.Coefficient of x^p.y^q.z^r.t^s in the expansion of (ax+by+cz+dt)ⁿ   (where p+q+r+s=n) is     n!/p!q!r!s! .  a^p.b^q.c^r.d^s.

 

 

 

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